Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Percentage Accurate: 93.6% → 99.3%

Time: 16.7s

Alternatives: 24

Speedup: 1.0×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x} - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8.6e-66)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma -0.5 (log x) 0.91893853320467))
   (fma
    (+ x -0.5)
    (log x)
    (+
     0.91893853320467
     (fma
      z
      (fma z (+ (/ 0.0007936500793651 x) (/ y x)) (/ -0.0027777777777778 x))
      (- (/ 0.083333333333333 x) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.6e-66) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	} else {
		tmp = fma((x + -0.5), log(x), (0.91893853320467 + fma(z, fma(z, ((0.0007936500793651 / x) + (y / x)), (-0.0027777777777778 / x)), ((0.083333333333333 / x) - x))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8.6e-66)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = fma(Float64(x + -0.5), log(x), Float64(0.91893853320467 + fma(z, fma(z, Float64(Float64(0.0007936500793651 / x) + Float64(y / x)), Float64(-0.0027777777777778 / x)), Float64(Float64(0.083333333333333 / x) - x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8.6e-66], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 + N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.60000000000000027e-66

   1. Initial program 99.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\color{blue}{y + \frac{7936500793651}{10000000000000000}}, z, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \color{blue}{\frac{-13888888888889}{5000000000000000}}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \color{blue}{\frac{1}{x}}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)}\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)}\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}}\right)\right) \]

      19. neg-lowering-neg.f64 99.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(-x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{-1}{2} \cdot \log x + \frac{91893853320467}{100000000000000}}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{x}\right)}\right)\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, -1 \cdot \log \left(\frac{1}{x}\right), \frac{91893853320467}{100000000000000}\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)}, \frac{91893853320467}{100000000000000}\right)\right) \]

      7. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right), \frac{91893853320467}{100000000000000}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\log x}, \frac{91893853320467}{100000000000000}\right)\right) \]

      9. log-lowering-log.f64 99.8

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \color{blue}{\log x}, 0.91893853320467\right)\right) \]

   7. Simplified 99.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)}\right) \]

   if 8.60000000000000027e-66 < x

   1. Initial program 92.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

   4. Applied egg-rr 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)\right) - x}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right) - x\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right) - x\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \left(\color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} - x\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} - x\right)\right)}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} - x\right)}\right) \]

   7. Simplified 99.6%

      \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{0.91893853320467 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x} - x\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot z}{x}, y, \log x \cdot \left(x + -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, \frac{1}{x \cdot 12.000000000000048}\right) + \left(0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -5e+36)
     (fma (/ (* z z) x) y (* (log x) (+ x -0.5)))
     (if (<= t_0 INFINITY)
       (+
        (fma (log x) (+ x -0.5) (/ 1.0 (* x 12.000000000000048)))
        (- 0.91893853320467 x))
       (fma
        z
        (fma z (+ (/ 0.0007936500793651 x) (/ y x)) (/ -0.0027777777777778 x))
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -5e+36) {
		tmp = fma(((z * z) / x), y, (log(x) * (x + -0.5)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(log(x), (x + -0.5), (1.0 / (x * 12.000000000000048))) + (0.91893853320467 - x);
	} else {
		tmp = fma(z, fma(z, ((0.0007936500793651 / x) + (y / x)), (-0.0027777777777778 / x)), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -5e+36)
		tmp = fma(Float64(Float64(z * z) / x), y, Float64(log(x) * Float64(x + -0.5)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(log(x), Float64(x + -0.5), Float64(1.0 / Float64(x * 12.000000000000048))) + Float64(0.91893853320467 - x));
	else
		tmp = fma(z, fma(z, Float64(Float64(0.0007936500793651 / x) + Float64(y / x)), Float64(-0.0027777777777778 / x)), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+36], N[(N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision] * y + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999977e36

   1. Initial program 89.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

   4. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{y \cdot {z}^{2}}{x}}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \color{blue}{\frac{{z}^{2}}{x}}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

      5. *-lowering-*.f64 88.0

         \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

   7. Simplified 88.0%

      \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{y \cdot \frac{z \cdot z}{x}}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{z \cdot z}{x} + \left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{x} \cdot y} + \left(x + \frac{-1}{2}\right) \cdot \log x \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot z}{x}, y, \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z \cdot z}{x}}, y, \left(x + \frac{-1}{2}\right) \cdot \log x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z \cdot z}}{x}, y, \left(x + \frac{-1}{2}\right) \cdot \log x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z \cdot z}{x}, y, \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z \cdot z}{x}, y, \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x\right) \]

      8. log-lowering-log.f64 88.0

         \[\leadsto \mathsf{fma}\left(\frac{z \cdot z}{x}, y, \left(x + -0.5\right) \cdot \color{blue}{\log x}\right) \]

   9. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot z}{x}, y, \left(x + -0.5\right) \cdot \log x\right)} \]

   if -4.99999999999999977e36 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 96.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      14. --lowering--.f64 67.3

          \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. metadata-eval 67.3

         \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{1}{x \cdot \color{blue}{12.000000000000048}}\right) + \left(0.91893853320467 - x\right) \]

   7. Applied egg-rr 67.3%

      \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \color{blue}{\frac{1}{x \cdot 12.000000000000048}}\right) + \left(0.91893853320467 - x\right) \]

   if +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 95.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 60.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}\right)}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}\right)}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{1} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      10. +-lowering-+.f64 28.1

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \color{blue}{\left(0.0007936500793651 + y\right)}, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x} \]

   10. Simplified 28.1%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \left(0.0007936500793651 + y\right), -0.0027777777777778\right), 0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \color{blue}{\frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       11. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}}}{x}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\frac{-13888888888889}{5000000000000000}}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) \]

       16. /-lowering-/.f64 57.9

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \color{blue}{\frac{0.083333333333333}{x}}\right) \]

   13. Simplified 57.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot z}{x}, y, \log x \cdot \left(x + -0.5\right)\right)\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\log x, x + -0.5, \frac{1}{x \cdot 12.000000000000048}\right) + \left(0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot z}{x}, y, \log x \cdot \left(x + -0.5\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -5e+36)
     (fma (/ (* z z) x) y (* (log x) (+ x -0.5)))
     (if (<= t_0 INFINITY)
       (fma
        (/ 1.0 x)
        0.083333333333333
        (- (fma (log x) (+ x -0.5) 0.91893853320467) x))
       (fma
        z
        (fma z (+ (/ 0.0007936500793651 x) (/ y x)) (/ -0.0027777777777778 x))
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -5e+36) {
		tmp = fma(((z * z) / x), y, (log(x) * (x + -0.5)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((1.0 / x), 0.083333333333333, (fma(log(x), (x + -0.5), 0.91893853320467) - x));
	} else {
		tmp = fma(z, fma(z, ((0.0007936500793651 / x) + (y / x)), (-0.0027777777777778 / x)), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -5e+36)
		tmp = fma(Float64(Float64(z * z) / x), y, Float64(log(x) * Float64(x + -0.5)));
	elseif (t_0 <= Inf)
		tmp = fma(Float64(1.0 / x), 0.083333333333333, Float64(fma(log(x), Float64(x + -0.5), 0.91893853320467) - x));
	else
		tmp = fma(z, fma(z, Float64(Float64(0.0007936500793651 / x) + Float64(y / x)), Float64(-0.0027777777777778 / x)), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+36], N[(N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision] * y + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999977e36

   1. Initial program 89.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

   4. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{y \cdot {z}^{2}}{x}}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \color{blue}{\frac{{z}^{2}}{x}}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

      5. *-lowering-*.f64 88.0

         \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

   7. Simplified 88.0%

      \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{y \cdot \frac{z \cdot z}{x}}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{z \cdot z}{x} + \left(x + \frac{-1}{2}\right) \cdot \log x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{x} \cdot y} + \left(x + \frac{-1}{2}\right) \cdot \log x \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot z}{x}, y, \left(x + \frac{-1}{2}\right) \cdot \log x\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z \cdot z}{x}}, y, \left(x + \frac{-1}{2}\right) \cdot \log x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z \cdot z}}{x}, y, \left(x + \frac{-1}{2}\right) \cdot \log x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z \cdot z}{x}, y, \color{blue}{\left(x + \frac{-1}{2}\right) \cdot \log x}\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z \cdot z}{x}, y, \color{blue}{\left(x + \frac{-1}{2}\right)} \cdot \log x\right) \]

      8. log-lowering-log.f64 88.0

         \[\leadsto \mathsf{fma}\left(\frac{z \cdot z}{x}, y, \left(x + -0.5\right) \cdot \color{blue}{\log x}\right) \]

   9. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z \cdot z}{x}, y, \left(x + -0.5\right) \cdot \log x\right)} \]

   if -4.99999999999999977e36 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 96.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      14. --lowering--.f64 67.3

          \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \log x \cdot \left(\frac{-1}{2} + x\right)\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \log x \cdot \color{blue}{\left(x + \frac{-1}{2}\right)}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \log x \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \log x \cdot \color{blue}{\left(x - \frac{1}{2}\right)}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(x - \frac{1}{2}\right) \cdot \log x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(x - \frac{1}{2}\right) \cdot \log x\right) + \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)} \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      9. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} + \left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} + \left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \frac{1}{x} \cdot \frac{83333333333333}{1000000000000000} + \left(\color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{x} \cdot \frac{83333333333333}{1000000000000000} + \left(\left(x + \color{blue}{\frac{-1}{2}}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \left(x + \frac{-1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{x}}, \frac{83333333333333}{1000000000000000}, \left(x + \frac{-1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \color{blue}{\left(x - \frac{1}{2}\right)} \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)}\right) \]

      19. associate-+r- N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \frac{91893853320467}{100000000000000}\right) - x}\right) \]

   7. Applied egg-rr 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\right)} \]

   if +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 95.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 60.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}\right)}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}\right)}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{1} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      10. +-lowering-+.f64 28.1

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \color{blue}{\left(0.0007936500793651 + y\right)}, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x} \]

   10. Simplified 28.1%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \left(0.0007936500793651 + y\right), -0.0027777777777778\right), 0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \color{blue}{\frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       11. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}}}{x}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\frac{-13888888888889}{5000000000000000}}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) \]

       16. /-lowering-/.f64 57.9

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \color{blue}{\frac{0.083333333333333}{x}}\right) \]

   13. Simplified 57.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot z}{x}, y, \log x \cdot \left(x + -0.5\right)\right)\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, y \cdot \frac{z \cdot z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -5e+36)
     (fma x (log x) (* y (/ (* z z) x)))
     (if (<= t_0 INFINITY)
       (fma
        (/ 1.0 x)
        0.083333333333333
        (- (fma (log x) (+ x -0.5) 0.91893853320467) x))
       (fma
        z
        (fma z (+ (/ 0.0007936500793651 x) (/ y x)) (/ -0.0027777777777778 x))
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -5e+36) {
		tmp = fma(x, log(x), (y * ((z * z) / x)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((1.0 / x), 0.083333333333333, (fma(log(x), (x + -0.5), 0.91893853320467) - x));
	} else {
		tmp = fma(z, fma(z, ((0.0007936500793651 / x) + (y / x)), (-0.0027777777777778 / x)), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -5e+36)
		tmp = fma(x, log(x), Float64(y * Float64(Float64(z * z) / x)));
	elseif (t_0 <= Inf)
		tmp = fma(Float64(1.0 / x), 0.083333333333333, Float64(fma(log(x), Float64(x + -0.5), 0.91893853320467) - x));
	else
		tmp = fma(z, fma(z, Float64(Float64(0.0007936500793651 / x) + Float64(y / x)), Float64(-0.0027777777777778 / x)), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+36], N[(x * N[Log[x], $MachinePrecision] + N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + N[(N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999977e36

   1. Initial program 89.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

   4. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{y \cdot {z}^{2}}{x}}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \color{blue}{\frac{{z}^{2}}{x}}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

      5. *-lowering-*.f64 88.0

         \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

   7. Simplified 88.0%

      \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{y \cdot \frac{z \cdot z}{x}}\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{x}, \log x, y \cdot \frac{z \cdot z}{x}\right) \]
   9. Step-by-step derivation

   10. Simplified 88.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{x}, \log x, y \cdot \frac{z \cdot z}{x}\right) \]

   if -4.99999999999999977e36 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 96.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      14. --lowering--.f64 67.3

          \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \log x \cdot \left(\frac{-1}{2} + x\right)\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \log x \cdot \color{blue}{\left(x + \frac{-1}{2}\right)}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \log x \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \log x \cdot \color{blue}{\left(x - \frac{1}{2}\right)}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(x - \frac{1}{2}\right) \cdot \log x\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(x - \frac{1}{2}\right) \cdot \log x\right) + \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)} \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      9. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} + \left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} + \left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \frac{1}{x} \cdot \frac{83333333333333}{1000000000000000} + \left(\color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{x} \cdot \frac{83333333333333}{1000000000000000} + \left(\left(x + \color{blue}{\frac{-1}{2}}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \left(x + \frac{-1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right)} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{x}}, \frac{83333333333333}{1000000000000000}, \left(x + \frac{-1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \color{blue}{\left(x - \frac{1}{2}\right)} \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\mathsf{neg}\left(x\right)\right)\right)}\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \left(x - \frac{1}{2}\right) \cdot \log x + \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)}\right) \]

      19. associate-+r- N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{x}, \frac{83333333333333}{1000000000000000}, \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \frac{91893853320467}{100000000000000}\right) - x}\right) \]

   7. Applied egg-rr 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\right)} \]

   if +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 95.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 60.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}\right)}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}\right)}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{1} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      10. +-lowering-+.f64 28.1

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \color{blue}{\left(0.0007936500793651 + y\right)}, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x} \]

   10. Simplified 28.1%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \left(0.0007936500793651 + y\right), -0.0027777777777778\right), 0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \color{blue}{\frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       11. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}}}{x}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\frac{-13888888888889}{5000000000000000}}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) \]

       16. /-lowering-/.f64 57.9

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \color{blue}{\frac{0.083333333333333}{x}}\right) \]

   13. Simplified 57.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, y \cdot \frac{z \cdot z}{x}\right)\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, \mathsf{fma}\left(\log x, x + -0.5, 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, y \cdot \frac{z \cdot z}{x}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -5e+36)
     (fma x (log x) (* y (/ (* z z) x)))
     (if (<= t_0 INFINITY)
       (fma
        (+ x -0.5)
        (log x)
        (- (+ 0.91893853320467 (/ 0.083333333333333 x)) x))
       (fma
        z
        (fma z (+ (/ 0.0007936500793651 x) (/ y x)) (/ -0.0027777777777778 x))
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -5e+36) {
		tmp = fma(x, log(x), (y * ((z * z) / x)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x + -0.5), log(x), ((0.91893853320467 + (0.083333333333333 / x)) - x));
	} else {
		tmp = fma(z, fma(z, ((0.0007936500793651 / x) + (y / x)), (-0.0027777777777778 / x)), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -5e+36)
		tmp = fma(x, log(x), Float64(y * Float64(Float64(z * z) / x)));
	elseif (t_0 <= Inf)
		tmp = fma(Float64(x + -0.5), log(x), Float64(Float64(0.91893853320467 + Float64(0.083333333333333 / x)) - x));
	else
		tmp = fma(z, fma(z, Float64(Float64(0.0007936500793651 / x) + Float64(y / x)), Float64(-0.0027777777777778 / x)), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+36], N[(x * N[Log[x], $MachinePrecision] + N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999977e36

   1. Initial program 89.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

   4. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{y \cdot {z}^{2}}{x}}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \color{blue}{\frac{{z}^{2}}{x}}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

      5. *-lowering-*.f64 88.0

         \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

   7. Simplified 88.0%

      \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{y \cdot \frac{z \cdot z}{x}}\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{x}, \log x, y \cdot \frac{z \cdot z}{x}\right) \]
   9. Step-by-step derivation

   10. Simplified 88.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{x}, \log x, y \cdot \frac{z \cdot z}{x}\right) \]

   if -4.99999999999999977e36 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 96.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      14. --lowering--.f64 67.3

          \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]
   6. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\log x \cdot \left(\frac{-1}{2} + x\right) + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \log x \cdot \color{blue}{\left(x + \frac{-1}{2}\right)} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \log x \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \log x \cdot \color{blue}{\left(x - \frac{1}{2}\right)} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \frac{-1}{2}}, \log x, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      11. associate-+r- N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x}\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      14. /-lowering-/.f64 67.3

          \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \left(\color{blue}{\frac{0.083333333333333}{x}} + 0.91893853320467\right) - x\right) \]

   7. Applied egg-rr 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)} \]

   if +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 95.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 60.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}\right)}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}\right)}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{1} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      10. +-lowering-+.f64 28.1

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \color{blue}{\left(0.0007936500793651 + y\right)}, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x} \]

   10. Simplified 28.1%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \left(0.0007936500793651 + y\right), -0.0027777777777778\right), 0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \color{blue}{\frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       11. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}}}{x}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\frac{-13888888888889}{5000000000000000}}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) \]

       16. /-lowering-/.f64 57.9

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \color{blue}{\frac{0.083333333333333}{x}}\right) \]

   13. Simplified 57.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, y \cdot \frac{z \cdot z}{x}\right)\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(y + 0.0007936500793651\right) \cdot \left(z \cdot z\right)}{x}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -5e+36)
     (/ (* (+ y 0.0007936500793651) (* z z)) x)
     (if (<= t_0 INFINITY)
       (fma
        (+ x -0.5)
        (log x)
        (- (+ 0.91893853320467 (/ 0.083333333333333 x)) x))
       (fma
        z
        (fma z (+ (/ 0.0007936500793651 x) (/ y x)) (/ -0.0027777777777778 x))
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -5e+36) {
		tmp = ((y + 0.0007936500793651) * (z * z)) / x;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x + -0.5), log(x), ((0.91893853320467 + (0.083333333333333 / x)) - x));
	} else {
		tmp = fma(z, fma(z, ((0.0007936500793651 / x) + (y / x)), (-0.0027777777777778 / x)), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -5e+36)
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) * Float64(z * z)) / x);
	elseif (t_0 <= Inf)
		tmp = fma(Float64(x + -0.5), log(x), Float64(Float64(0.91893853320467 + Float64(0.083333333333333 / x)) - x));
	else
		tmp = fma(z, fma(z, Float64(Float64(0.0007936500793651 / x) + Float64(y / x)), Float64(-0.0027777777777778 / x)), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+36], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999977e36

   1. Initial program 89.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 25.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 28.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}}{x} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}}{x} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}}{x} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot {z}^{2}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \color{blue}{\left(z \cdot z\right)}}{x} \]

      5. *-lowering-*.f64 87.6

         \[\leadsto \frac{\left(0.0007936500793651 + y\right) \cdot \color{blue}{\left(z \cdot z\right)}}{x} \]

   10. Simplified 87.6%

       \[\leadsto \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot \left(z \cdot z\right)}}{x} \]

   if -4.99999999999999977e36 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 96.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      14. --lowering--.f64 67.3

          \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]
   6. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\log x \cdot \left(\frac{-1}{2} + x\right) + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \log x \cdot \color{blue}{\left(x + \frac{-1}{2}\right)} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \log x \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \log x \cdot \color{blue}{\left(x - \frac{1}{2}\right)} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} + \left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \frac{-1}{2}}, \log x, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \frac{\frac{83333333333333}{1000000000000000}}{x} + \left(\frac{91893853320467}{100000000000000} - x\right)\right) \]

      11. associate-+r- N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x}\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right) - x}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      14. /-lowering-/.f64 67.3

          \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \left(\color{blue}{\frac{0.083333333333333}{x}} + 0.91893853320467\right) - x\right) \]

   7. Applied egg-rr 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)} \]

   if +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 95.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 60.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}\right)}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}\right)}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{1} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      10. +-lowering-+.f64 28.1

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \color{blue}{\left(0.0007936500793651 + y\right)}, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x} \]

   10. Simplified 28.1%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \left(0.0007936500793651 + y\right), -0.0027777777777778\right), 0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \color{blue}{\frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       11. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}}}{x}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\frac{-13888888888889}{5000000000000000}}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) \]

       16. /-lowering-/.f64 57.9

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \color{blue}{\frac{0.083333333333333}{x}}\right) \]

   13. Simplified 57.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(y + 0.0007936500793651\right) \cdot \left(z \cdot z\right)}{x}\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{0.083333333333333}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(y + 0.0007936500793651\right) \cdot \left(z \cdot z\right)}{x}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(0.91893853320467 - x\right) + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
          (/
           (+
            0.083333333333333
            (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
           x))))
   (if (<= t_0 -5e+36)
     (/ (* (+ y 0.0007936500793651) (* z z)) x)
     (if (<= t_0 INFINITY)
       (+
        (- 0.91893853320467 x)
        (fma (log x) (+ x -0.5) (/ 0.083333333333333 x)))
       (fma
        z
        (fma z (+ (/ 0.0007936500793651 x) (/ y x)) (/ -0.0027777777777778 x))
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	double tmp;
	if (t_0 <= -5e+36) {
		tmp = ((y + 0.0007936500793651) * (z * z)) / x;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (0.91893853320467 - x) + fma(log(x), (x + -0.5), (0.083333333333333 / x));
	} else {
		tmp = fma(z, fma(z, ((0.0007936500793651 / x) + (y / x)), (-0.0027777777777778 / x)), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x))
	tmp = 0.0
	if (t_0 <= -5e+36)
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) * Float64(z * z)) / x);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(0.91893853320467 - x) + fma(log(x), Float64(x + -0.5), Float64(0.083333333333333 / x)));
	else
		tmp = fma(z, fma(z, Float64(Float64(0.0007936500793651 / x) + Float64(y / x)), Float64(-0.0027777777777778 / x)), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+36], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(0.91893853320467 - x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x + -0.5), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999977e36

   1. Initial program 89.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 25.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 28.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}}{x} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}}{x} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}}{x} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot {z}^{2}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \color{blue}{\left(z \cdot z\right)}}{x} \]

      5. *-lowering-*.f64 87.6

         \[\leadsto \frac{\left(0.0007936500793651 + y\right) \cdot \color{blue}{\left(z \cdot z\right)}}{x} \]

   10. Simplified 87.6%

       \[\leadsto \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot \left(z \cdot z\right)}}{x} \]

   if -4.99999999999999977e36 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < +inf.0

   1. Initial program 96.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      14. --lowering--.f64 67.3

          \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

   5. Simplified 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]

   if +inf.0 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 95.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 60.3%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{z \cdot \left(z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}\right)}{x} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}\right)}{x} \]

      3. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, z \cdot \left(\frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} - -1 \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{1} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{192901234567904320987654321}{2083333333333325000000000000000} + \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right) \cdot \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{192901234567904320987654321}{2083333333333325000000000000000} + \left(\frac{7936500793651}{10000000000000000} + y\right)}, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right)\right)}{x} \]

      10. +-lowering-+.f64 28.1

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \color{blue}{\left(0.0007936500793651 + y\right)}, -0.0027777777777778\right), 0.083333333333333\right)\right)}{x} \]

   10. Simplified 28.1%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 9.259259259259444 \cdot 10^{-5} + \left(0.0007936500793651 + y\right), -0.0027777777777778\right), 0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}} \]
   12. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

       2. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} + \frac{y}{x}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \color{blue}{\frac{y}{x}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\color{blue}{\frac{\frac{13888888888889}{5000000000000000} \cdot 1}{x}}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \mathsf{neg}\left(\frac{\color{blue}{\frac{13888888888889}{5000000000000000}}}{x}\right)\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       11. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\color{blue}{\frac{-13888888888889}{5000000000000000}}}{x}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \color{blue}{\frac{\frac{-13888888888889}{5000000000000000}}{x}}\right), \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) \]

       14. associate-*r/ N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{7936500793651}{10000000000000000}}{x} + \frac{y}{x}, \frac{\frac{-13888888888889}{5000000000000000}}{x}\right), \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) \]

       16. /-lowering-/.f64 57.9

           \[\leadsto \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \color{blue}{\frac{0.083333333333333}{x}}\right) \]

   13. Simplified 57.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\left(y + 0.0007936500793651\right) \cdot \left(z \cdot z\right)}{x}\\ \mathbf{elif}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq \infty:\\ \;\;\;\;\left(0.91893853320467 - x\right) + \mathsf{fma}\left(\log x, x + -0.5, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 + \mathsf{fma}\left(z, z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right), \frac{0.083333333333333}{x} - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e-21)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma -0.5 (log x) 0.91893853320467))
   (fma
    (+ x -0.5)
    (log x)
    (+
     0.91893853320467
     (fma
      z
      (* z (+ (/ 0.0007936500793651 x) (/ y x)))
      (- (/ 0.083333333333333 x) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e-21) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	} else {
		tmp = fma((x + -0.5), log(x), (0.91893853320467 + fma(z, (z * ((0.0007936500793651 / x) + (y / x))), ((0.083333333333333 / x) - x))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e-21)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = fma(Float64(x + -0.5), log(x), Float64(0.91893853320467 + fma(z, Float64(z * Float64(Float64(0.0007936500793651 / x) + Float64(y / x))), Float64(Float64(0.083333333333333 / x) - x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e-21], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 + N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999963e-21

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\color{blue}{y + \frac{7936500793651}{10000000000000000}}, z, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \color{blue}{\frac{-13888888888889}{5000000000000000}}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \color{blue}{\frac{1}{x}}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)}\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)}\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}}\right)\right) \]

      19. neg-lowering-neg.f64 99.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(-x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{-1}{2} \cdot \log x + \frac{91893853320467}{100000000000000}}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{x}\right)}\right)\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, -1 \cdot \log \left(\frac{1}{x}\right), \frac{91893853320467}{100000000000000}\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)}, \frac{91893853320467}{100000000000000}\right)\right) \]

      7. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right), \frac{91893853320467}{100000000000000}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\log x}, \frac{91893853320467}{100000000000000}\right)\right) \]

      9. log-lowering-log.f64 99.7

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \color{blue}{\log x}, 0.91893853320467\right)\right) \]

   7. Simplified 99.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)}\right) \]

   if 3.99999999999999963e-21 < x

   1. Initial program 91.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

   4. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)\right) - x}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right) - x\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right) - x\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \left(\color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} - x\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} - x\right)\right)}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} - x\right)}\right) \]

   7. Simplified 99.7%

      \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{0.91893853320467 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x} - x\right)}\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, z \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}\right), \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, z \cdot \left(\frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} + \frac{y}{x}\right), \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} + \frac{y}{x}\right), \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

      6. /-lowering-/.f64 99.7

         \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 + \mathsf{fma}\left(z, z \cdot \left(\frac{0.0007936500793651}{x} + \color{blue}{\frac{y}{x}}\right), \frac{0.083333333333333}{x} - x\right)\right) \]

   10. Simplified 99.7%

       \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 + \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)}, \frac{0.083333333333333}{x} - x\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5.2e+149)
   (fma
    (+ x -0.5)
    (log x)
    (-
     (+
      0.91893853320467
      (/
       (fma
        z
        (fma (+ y 0.0007936500793651) z -0.0027777777777778)
        0.083333333333333)
       x))
     x))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5.2e+149) {
		tmp = fma((x + -0.5), log(x), ((0.91893853320467 + (fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x)) - x));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((z * (y + 0.0007936500793651)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5.2e+149)
		tmp = fma(Float64(x + -0.5), log(x), Float64(Float64(0.91893853320467 + Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x)) - x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5.2e+149], N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 + N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.19999999999999957e149

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

   if 5.19999999999999957e149 < x

   1. Initial program 84.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(z \cdot z\right)} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{z \cdot \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + z \cdot \color{blue}{\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + z \cdot \color{blue}{\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + z \cdot \frac{\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}}{x} \]

      8. +-lowering-+.f64 97.1

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + z \cdot \frac{z \cdot \color{blue}{\left(0.0007936500793651 + y\right)}}{x} \]

   5. Simplified 97.1%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.2)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma -0.5 (log x) 0.91893853320467))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (/ (* z (+ y 0.0007936500793651)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.2) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((z * (y + 0.0007936500793651)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.2)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.2], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.20000000000000001

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\color{blue}{y + \frac{7936500793651}{10000000000000000}}, z, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \color{blue}{\frac{-13888888888889}{5000000000000000}}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \color{blue}{\frac{1}{x}}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)}\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)}\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}}\right)\right) \]

      19. neg-lowering-neg.f64 99.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(-x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{-1}{2} \cdot \log x + \frac{91893853320467}{100000000000000}}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{x}\right)}\right)\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, -1 \cdot \log \left(\frac{1}{x}\right), \frac{91893853320467}{100000000000000}\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)}, \frac{91893853320467}{100000000000000}\right)\right) \]

      7. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right), \frac{91893853320467}{100000000000000}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\log x}, \frac{91893853320467}{100000000000000}\right)\right) \]

      9. log-lowering-log.f64 99.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \color{blue}{\log x}, 0.91893853320467\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)}\right) \]

   if 0.20000000000000001 < x

   1. Initial program 91.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(z \cdot z\right)} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{z \cdot \left(z \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + z \cdot \color{blue}{\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + z \cdot \color{blue}{\frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + z \cdot \frac{\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}}{x} \]

      8. +-lowering-+.f64 97.1

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + z \cdot \frac{z \cdot \color{blue}{\left(0.0007936500793651 + y\right)}}{x} \]

   5. Simplified 97.1%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.26:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{z \cdot \left(z \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.26)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma -0.5 (log x) 0.91893853320467))
   (+ (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)) (/ (* z (* z y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.26) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((z * (z * y)) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.26)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(-0.5, log(x), 0.91893853320467));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(z * Float64(z * y)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.26], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.26000000000000001

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right)} + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, \left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\color{blue}{y + \frac{7936500793651}{10000000000000000}}, z, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \color{blue}{\frac{-13888888888889}{5000000000000000}}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \color{blue}{\frac{1}{x}}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)}\right) \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)}\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}}\right)\right) \]

      19. neg-lowering-neg.f64 99.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(-x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{91893853320467}{100000000000000} + \frac{-1}{2} \cdot \log x}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\frac{-1}{2} \cdot \log x + \frac{91893853320467}{100000000000000}}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{x}\right)}\right)\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \frac{-1}{2} \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)} + \frac{91893853320467}{100000000000000}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, -1 \cdot \log \left(\frac{1}{x}\right), \frac{91893853320467}{100000000000000}\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)}, \frac{91893853320467}{100000000000000}\right)\right) \]

      7. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right), \frac{91893853320467}{100000000000000}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\log x}, \frac{91893853320467}{100000000000000}\right)\right) \]

      9. log-lowering-log.f64 99.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \color{blue}{\log x}, 0.91893853320467\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)}\right) \]

   if 0.26000000000000001 < x

   1. Initial program 91.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{y \cdot {z}^{2}}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{x} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{z \cdot \left(z \cdot y\right)}}{x} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{z \cdot \color{blue}{\left(y \cdot z\right)}}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{z \cdot \left(y \cdot z\right)}}{x} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{z \cdot \color{blue}{\left(z \cdot y\right)}}{x} \]

      7. *-lowering-*.f64 87.8

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{z \cdot \color{blue}{\left(z \cdot y\right)}}{x} \]

   5. Simplified 87.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{z \cdot \left(z \cdot y\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.26:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{z \cdot \left(z \cdot y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.26:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{z \cdot \left(z \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.26)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (+ (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)) (/ (* z (* z y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.26) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((z * (z * y)) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.26)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(z * Float64(z * y)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.26], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.26000000000000001

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. +-lowering-+.f64 99.2

         \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{0.0007936500793651 + y}, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

   if 0.26000000000000001 < x

   1. Initial program 91.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{y \cdot {z}^{2}}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{x} \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{z \cdot \left(z \cdot y\right)}}{x} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{z \cdot \color{blue}{\left(y \cdot z\right)}}{x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{z \cdot \left(y \cdot z\right)}}{x} \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{z \cdot \color{blue}{\left(z \cdot y\right)}}{x} \]

      7. *-lowering-*.f64 87.8

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{z \cdot \color{blue}{\left(z \cdot y\right)}}{x} \]

   5. Simplified 87.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{z \cdot \left(z \cdot y\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.26:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{z \cdot \left(z \cdot y\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.5e+84)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma x (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.5e+84) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma(x, log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.5e+84)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(x, log(x), Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.5e+84], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.50000000000000027e84

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. +-lowering-+.f64 89.3

         \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{0.0007936500793651 + y}, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

   if 6.50000000000000027e84 < x

   1. Initial program 87.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      3. log-rec N/A

         \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      4. remove-double-neg N/A

         \[\leadsto x \cdot \left(\color{blue}{\log x} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto x \cdot \left(\log x + \color{blue}{-1}\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\log x \cdot x + -1 \cdot x} \]

      7. neg-mul-1 N/A

         \[\leadsto \log x \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \log x} + \left(\mathsf{neg}\left(x\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x, \mathsf{neg}\left(x\right)\right)} \]

      10. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log x}, \mathsf{neg}\left(x\right)\right) \]

      11. neg-lowering-neg.f64 83.3

          \[\leadsto \mathsf{fma}\left(x, \log x, \color{blue}{-x}\right) \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x, -x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log x, -x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.75e+84)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (* x (log x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.75e+84) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = x * log(x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.75e+84)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(x * log(x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.75e+84], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7499999999999999e84

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. +-lowering-+.f64 89.3

         \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{0.0007936500793651 + y}, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

   if 1.7499999999999999e84 < x

   1. Initial program 87.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

   4. Applied egg-rr 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{y \cdot {z}^{2}}{x}}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \color{blue}{\frac{{z}^{2}}{x}}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

      5. *-lowering-*.f64 38.3

         \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, y \cdot \frac{\color{blue}{z \cdot z}}{x}\right) \]

   7. Simplified 38.3%

      \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{y \cdot \frac{z \cdot z}{x}}\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \log \left(\frac{1}{x}\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \log \left(\frac{1}{x}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} \]

      3. log-rec N/A

         \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) \]

      4. remove-double-neg N/A

         \[\leadsto x \cdot \color{blue}{\log x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log x} \]

      6. log-lowering-log.f64 27.7

         \[\leadsto x \cdot \color{blue}{\log x} \]

   10. Simplified 27.7%

       \[\leadsto \color{blue}{x \cdot \log x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ t_1 := y \cdot \frac{z \cdot z}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+19}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
        (t_1 (* y (/ (* z z) x))))
   (if (<= t_0 -2e+49)
     t_1
     (if (<= t_0 1e+19) (/ 1.0 (* x 12.000000000000048)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = y * ((z * z) / x);
	double tmp;
	if (t_0 <= -2e+49) {
		tmp = t_1;
	} else if (t_0 <= 1e+19) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)
    t_1 = y * ((z * z) / x)
    if (t_0 <= (-2d+49)) then
        tmp = t_1
    else if (t_0 <= 1d+19) then
        tmp = 1.0d0 / (x * 12.000000000000048d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = y * ((z * z) / x);
	double tmp;
	if (t_0 <= -2e+49) {
		tmp = t_1;
	} else if (t_0 <= 1e+19) {
		tmp = 1.0 / (x * 12.000000000000048);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)
	t_1 = y * ((z * z) / x)
	tmp = 0
	if t_0 <= -2e+49:
		tmp = t_1
	elif t_0 <= 1e+19:
		tmp = 1.0 / (x * 12.000000000000048)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	t_1 = Float64(y * Float64(Float64(z * z) / x))
	tmp = 0.0
	if (t_0 <= -2e+49)
		tmp = t_1;
	elseif (t_0 <= 1e+19)
		tmp = Float64(1.0 / Float64(x * 12.000000000000048));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	t_1 = y * ((z * z) / x);
	tmp = 0.0;
	if (t_0 <= -2e+49)
		tmp = t_1;
	elseif (t_0 <= 1e+19)
		tmp = 1.0 / (x * 12.000000000000048);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+49], t$95$1, If[LessEqual[t$95$0, 1e+19], N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1.99999999999999989e49 or 1e19 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 92.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{{z}^{2}}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{{z}^{2}}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{{z}^{2}}{x}} \]

      4. unpow2 N/A

         \[\leadsto y \cdot \frac{\color{blue}{z \cdot z}}{x} \]

      5. *-lowering-*.f64 55.6

         \[\leadsto y \cdot \frac{\color{blue}{z \cdot z}}{x} \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{y \cdot \frac{z \cdot z}{x}} \]

   if -1.99999999999999989e49 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1e19

   1. Initial program 99.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

      14. --lowering--.f64 97.5

          \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

   5. Simplified 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 47.1

         \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

   8. Simplified 47.1%

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

      3. div-inv N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

      5. metadata-eval 47.2

         \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

   10. Applied egg-rr 47.2%

       \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -2 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 10^{+19}:\\ \;\;\;\;\frac{1}{x \cdot 12.000000000000048}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + 0.0007936500793651\right) \cdot \left(z \cdot z\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)) 5.0)
   (/ (fma (* z z) y 0.083333333333333) x)
   (/ (* (+ y 0.0007936500793651) (* z z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) <= 5.0) {
		tmp = fma((z * z), y, 0.083333333333333) / x;
	} else {
		tmp = ((y + 0.0007936500793651) * (z * z)) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) <= 5.0)
		tmp = Float64(fma(Float64(z * z), y, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) * Float64(z * z)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[(N[(z * z), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5

   1. Initial program 97.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 79.2%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 42.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{y}, \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

   10. Simplified 55.3%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{y}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \color{blue}{\frac{83333333333333}{1000000000000000}}\right)}{x} \]
   12. Step-by-step derivation

   13. Simplified 55.5%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \color{blue}{0.083333333333333}\right)}{x} \]

   if 5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 92.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 33.1%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 20.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}}{x} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}}{x} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}}{x} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot {z}^{2}}{x} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \color{blue}{\left(z \cdot z\right)}}{x} \]

      5. *-lowering-*.f64 69.3

         \[\leadsto \frac{\left(0.0007936500793651 + y\right) \cdot \color{blue}{\left(z \cdot z\right)}}{x} \]

   10. Simplified 69.3%

       \[\leadsto \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot \left(z \cdot z\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + 0.0007936500793651\right) \cdot \left(z \cdot z\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{y + 0.0007936500793651}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)) 4e+24)
   (/ (fma (* z z) y 0.083333333333333) x)
   (* (* z z) (/ (+ y 0.0007936500793651) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) <= 4e+24) {
		tmp = fma((z * z), y, 0.083333333333333) / x;
	} else {
		tmp = (z * z) * ((y + 0.0007936500793651) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) <= 4e+24)
		tmp = Float64(fma(Float64(z * z), y, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(z * z) * Float64(Float64(y + 0.0007936500793651) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision], 4e+24], N[(N[(N[(z * z), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 3.9999999999999999e24

   1. Initial program 97.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 79.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 42.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{y}, \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

   10. Simplified 54.2%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{y}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \color{blue}{\frac{83333333333333}{1000000000000000}}\right)}{x} \]
   12. Step-by-step derivation

   13. Simplified 54.6%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \color{blue}{0.083333333333333}\right)}{x} \]

   if 3.9999999999999999e24 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 92.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 29.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 19.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      3. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(z \cdot z\right) \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      6. +-lowering-+.f64 69.6

         \[\leadsto \left(z \cdot z\right) \cdot \frac{\color{blue}{0.0007936500793651 + y}}{x} \]

   10. Simplified 69.6%

       \[\leadsto \color{blue}{\left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{y + 0.0007936500793651}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(z \cdot z, y, 0.083333333333333\right)}{x}\\ \mathbf{if}\;y \leq -0.0008:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (fma (* z z) y 0.083333333333333) x)))
   (if (<= y -0.0008)
     t_0
     (if (<= y -2.25e-302)
       (/ (fma -0.0027777777777778 z 0.083333333333333) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z * z), y, 0.083333333333333) / x;
	double tmp;
	if (y <= -0.0008) {
		tmp = t_0;
	} else if (y <= -2.25e-302) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(Float64(z * z), y, 0.083333333333333) / x)
	tmp = 0.0
	if (y <= -0.0008)
		tmp = t_0;
	elseif (y <= -2.25e-302)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z * z), $MachinePrecision] * y + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -0.0008], t$95$0, If[LessEqual[y, -2.25e-302], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000038e-4 or -2.25000000000000005e-302 < y

   1. Initial program 95.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 55.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 30.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{y}, \frac{\frac{-6944444444444388888888888889}{1000000000000000000000000000000}}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), \frac{-83333333333333}{1000000000000000}\right)}\right)}{x} \]
   9. Step-by-step derivation

   10. Simplified 59.8%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, \color{blue}{y}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x} \]

   11. Taylor expanded in z around 0

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \color{blue}{\frac{83333333333333}{1000000000000000}}\right)}{x} \]
   12. Step-by-step derivation

   13. Simplified 60.0%

       \[\leadsto \frac{\mathsf{fma}\left(z \cdot z, y, \color{blue}{0.083333333333333}\right)}{x} \]

   if -8.00000000000000038e-4 < y < -2.25000000000000005e-302

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

      2. flip-+ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

      3. associate-*l/ N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. Applied egg-rr 75.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

   7. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}}{x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

      2. accelerator-lowering-fma.f64 42.8

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}}{x} \]

   10. Simplified 42.8%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 64.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (fma
   z
   (fma z (+ y 0.0007936500793651) -0.0027777777777778)
   0.083333333333333)
  x))

C

double code(double x, double y, double z) {
	return fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

   4. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \color{blue}{\frac{-13888888888889}{5000000000000000}}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right)}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

   7. +-lowering-+.f64 62.3

      \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{0.0007936500793651 + y}, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]

5. Simplified 62.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]

6. Final simplification 62.3%

   \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
7. Add Preprocessing

Alternative 20: 63.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(y + 0.0007936500793651, z \cdot z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (fma (+ y 0.0007936500793651) (* z z) 0.083333333333333) x))

C

double code(double x, double y, double z) {
	return fma((y + 0.0007936500793651), (z * z), 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(Float64(y + 0.0007936500793651), Float64(z * z), 0.083333333333333) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right)} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x + \color{blue}{\frac{-1}{2}}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \color{blue}{\log x}, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

   10. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)}\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \left(\mathsf{neg}\left(x\right)\right) + \left(\frac{91893853320467}{100000000000000} + \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}}\right)\right) \]

4. Applied egg-rr 95.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x + -0.5, \log x, \left(-x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\right)} \]

5. Taylor expanded in z around 0

   \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right)\right) - x}\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right) - x\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\frac{91893853320467}{100000000000000} + \left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right)\right) - x\right)}\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \left(\color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} - x\right)\right) \]

   4. associate--l+ N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} - x\right)\right)}\right) \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} - x\right)}\right) \]

7. Simplified 94.2%

   \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{0.91893853320467 + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.0007936500793651}{x} + \frac{y}{x}, \frac{-0.0027777777777778}{x}\right), \frac{0.083333333333333}{x} - x\right)}\right) \]

8. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, z \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}\right), \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, z \cdot \left(\frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} + \frac{y}{x}\right), \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \frac{91893853320467}{100000000000000} + \mathsf{fma}\left(z, z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000}}{x}} + \frac{y}{x}\right), \frac{\frac{83333333333333}{1000000000000000}}{x} - x\right)\right) \]

   6. /-lowering-/.f64 93.8

      \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 + \mathsf{fma}\left(z, z \cdot \left(\frac{0.0007936500793651}{x} + \color{blue}{\frac{y}{x}}\right), \frac{0.083333333333333}{x} - x\right)\right) \]

10. Simplified 93.8%

    \[\leadsto \mathsf{fma}\left(x + -0.5, \log x, 0.91893853320467 + \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)}, \frac{0.083333333333333}{x} - x\right)\right) \]

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
12. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + {z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]

    2. +-commutative N/A

       \[\leadsto \frac{\color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

    3. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}} + \frac{83333333333333}{1000000000000000}}{x} \]

    4. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, {z}^{2}, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{7936500793651}{10000000000000000} + y}, {z}^{2}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

    6. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, \color{blue}{z \cdot z}, \frac{83333333333333}{1000000000000000}\right)}{x} \]

    7. *-lowering-*.f64 61.6

       \[\leadsto \frac{\mathsf{fma}\left(0.0007936500793651 + y, \color{blue}{z \cdot z}, 0.083333333333333\right)}{x} \]

13. Simplified 61.6%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0007936500793651 + y, z \cdot z, 0.083333333333333\right)}{x}} \]

14. Final simplification 61.6%

    \[\leadsto \frac{\mathsf{fma}\left(y + 0.0007936500793651, z \cdot z, 0.083333333333333\right)}{x} \]
15. Add Preprocessing

Alternative 21: 29.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (fma -0.0027777777777778 z 0.083333333333333) x))

C

double code(double x, double y, double z) {
	return fma(-0.0027777777777778, z, 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} \]

   2. flip-+ N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \cdot \frac{1}{x} \]

   3. associate-*l/ N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{\left(\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) \cdot \left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z\right) - \frac{83333333333333}{1000000000000000} \cdot \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z - \frac{83333333333333}{1000000000000000}}} \]

4. Applied egg-rr 60.3%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right) \cdot \left(z \cdot \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right)\right), -0.0069444444444443885\right) \cdot \frac{1}{x}}{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), -0.083333333333333\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{{z}^{2} \cdot {\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}^{2}}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}} - \frac{6944444444444388888888888889}{1000000000000000000000000000000} \cdot \frac{1}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) - \frac{83333333333333}{1000000000000000}}}{x}} \]

7. Simplified 33.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, \frac{\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right) \cdot \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}, \frac{-0.0069444444444443885}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), -0.083333333333333\right)}\right)}{x}} \]

8. Taylor expanded in z around 0

   \[\leadsto \frac{\color{blue}{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}}{x} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-13888888888889}{5000000000000000} \cdot z + \frac{83333333333333}{1000000000000000}}}{x} \]

   2. accelerator-lowering-fma.f64 30.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}}{x} \]

10. Simplified 30.1%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}}{x} \]
11. Add Preprocessing

Alternative 22: 24.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot 12.000000000000048} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (* x 12.000000000000048)))

C

double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (x * 12.000000000000048d0)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / (x * 12.000000000000048);
}

Python

def code(x, y, z):
	return 1.0 / (x * 12.000000000000048)

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(x * 12.000000000000048))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / (x * 12.000000000000048);
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   14. --lowering--.f64 59.1

       \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

5. Simplified 59.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 22.7

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

8. Simplified 22.7%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

   3. div-inv N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{1}{\frac{83333333333333}{1000000000000000}}}} \]

   5. metadata-eval 22.7

      \[\leadsto \frac{1}{x \cdot \color{blue}{12.000000000000048}} \]

10. Applied egg-rr 22.7%

    \[\leadsto \color{blue}{\frac{1}{x \cdot 12.000000000000048}} \]
11. Add Preprocessing

Alternative 23: 24.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ 0.083333333333333 \cdot \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.083333333333333 (/ 1.0 x)))

C

double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 * (1.0d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / x);
}

Python

def code(x, y, z):
	return 0.083333333333333 * (1.0 / x)

Julia

function code(x, y, z)
	return Float64(0.083333333333333 * Float64(1.0 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 * (1.0 / x);
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   14. --lowering--.f64 59.1

       \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

5. Simplified 59.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 22.7

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

8. Simplified 22.7%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} \]

   4. /-lowering-/.f64 22.7

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot 0.083333333333333 \]

10. Applied egg-rr 22.7%

    \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.083333333333333} \]

11. Final simplification 22.7%

    \[\leadsto 0.083333333333333 \cdot \frac{1}{x} \]
12. Add Preprocessing

Alternative 24: 24.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \frac{91893853320467}{100000000000000}\right)} - x \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right) + \left(\frac{91893853320467}{100000000000000} - x\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log x}, x - \frac{1}{2}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\log x, x + \color{blue}{\frac{-1}{2}}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log x, \color{blue}{\frac{-1}{2} + x}, \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log x, \frac{-1}{2} + x, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) + \left(\frac{91893853320467}{100000000000000} - x\right) \]

   14. --lowering--.f64 59.1

       \[\leadsto \mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \color{blue}{\left(0.91893853320467 - x\right)} \]

5. Simplified 59.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) + \left(0.91893853320467 - x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 22.7

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

8. Simplified 22.7%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
9. Add Preprocessing

Developer Target 1: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x))
  (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) + (0.91893853320467d0 - x)) + (0.083333333333333d0 / x)) + ((z / x) * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) + Float64(0.91893853320467 - x)) + Float64(0.083333333333333 / x)) + Float64(Float64(z / x) * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (* (- x 1/2) (log x)) (- 91893853320467/100000000000000 x)) (/ 83333333333333/1000000000000000 x)) (* (/ z x) (- (* z (+ y 7936500793651/10000000000000000)) 13888888888889/5000000000000000))))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))