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

Percentage Accurate: 94.1% → 98.5%

Time: 16.9s

Alternatives: 17

Speedup: 1.1×

• 31.4% 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: 94.1% 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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5e+36)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (fma (+ x -0.5) (log x) (- 0.91893853320467 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 <= 5e+36) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), fma((x + -0.5), log(x), (0.91893853320467 - 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 <= 5e+36)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), fma(Float64(x + -0.5), log(x), Float64(0.91893853320467 - 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, 5e+36], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $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 < 4.99999999999999977e36

   1. Initial program 99.6%

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

      1. +-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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(0 - x\right)} + \frac{91893853320467}{100000000000000}\right)\right) \]

      20. --lowering--.f64 99.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(0 - x\right) + 0.91893853320467\right)\right)} \]

   if 4.99999999999999977e36 < x

   1. Initial program 86.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 97.2

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + z \cdot \frac{z \cdot \color{blue}{\left(0.0007936500793651 + y\right)}}{x} \]

   5. Simplified 97.2%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

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

Alternative 2: 59.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -5 \cdot 10^{+55}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
       (/
        (+
         0.083333333333333
         (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
        x))
      -5e+55)
   (* (* z y) (/ z x))
   (/
    (fma z (fma z 0.0007936500793651 -0.0027777777777778) 0.083333333333333)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x)) <= -5e+55) {
		tmp = (z * y) * (z / x);
	} else {
		tmp = fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (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)) <= -5e+55)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	else
		tmp = Float64(fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[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], -5e+55], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 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)) < -5.00000000000000046e55

   1. Initial program 83.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

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

         \[\leadsto \frac{\color{blue}{y \cdot {z}^{2}}}{x} \]

      3. unpow2 N/A

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

      4. *-lowering-*.f64 84.0

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

   5. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 89.3

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

   7. Applied egg-rr 89.3%

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

   if -5.00000000000000046e55 < (+.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 95.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

   5. Simplified 93.6%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 54.2

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

   8. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} \leq -5 \cdot 10^{+55}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, x \cdot \left(\log x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.8e-19)
   (fma
    (fma
     z
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     0.083333333333333)
    (/ 1.0 x)
    (* x (+ (log x) -1.0)))
   (+
    (+ 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 <= 1.8e-19) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), (x * (log(x) + -1.0)));
	} 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 <= 1.8e-19)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), Float64(x * Float64(log(x) + -1.0)));
	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, 1.8e-19], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $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 < 1.8000000000000001e-19

   1. Initial program 99.6%

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

      1. +-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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(0 - x\right)} + \frac{91893853320467}{100000000000000}\right)\right) \]

      20. --lowering--.f64 99.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(0 - x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. remove-double-neg N/A

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

      17. log-lowering-log.f64 99.7

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

   7. Simplified 99.7%

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

   if 1.8000000000000001e-19 < x

   1. Initial program 88.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 97.4

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + z \cdot \frac{z \cdot \color{blue}{\left(0.0007936500793651 + y\right)}}{x} \]

   5. Simplified 97.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{z \cdot \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, x \cdot \left(\log x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z \cdot \left(y + 0.0007936500793651\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x + -1\\ \mathbf{if}\;x \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, x \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (log x) -1.0)))
   (if (<= x 100000000000.0)
     (fma
      (fma
       z
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       0.083333333333333)
      (/ 1.0 x)
      (* x t_0))
     (fma t_0 x (* (+ y 0.0007936500793651) (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) + -1.0;
	double tmp;
	if (x <= 100000000000.0) {
		tmp = fma(fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), (1.0 / x), (x * t_0));
	} else {
		tmp = fma(t_0, x, ((y + 0.0007936500793651) * ((z * z) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(x) + -1.0)
	tmp = 0.0
	if (x <= 100000000000.0)
		tmp = fma(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333), Float64(1.0 / x), Float64(x * t_0));
	else
		tmp = fma(t_0, x, Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[x, 100000000000.0], N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * x + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e11

   1. Initial program 99.6%

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

      1. +-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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(0 - x\right)} + \frac{91893853320467}{100000000000000}\right)\right) \]

      20. --lowering--.f64 99.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(0 - x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. remove-double-neg N/A

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

      17. log-lowering-log.f64 99.5

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

   7. Simplified 99.5%

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

   if 1e11 < x

   1. Initial program 87.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(0 - x\right)} + \frac{91893853320467}{100000000000000}\right)\right) \]

      20. --lowering--.f64 87.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(0 - x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. remove-double-neg N/A

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

      17. log-lowering-log.f64 87.8

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

   7. Simplified 87.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{x \cdot \left(-1 + \log x\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-lowering-+.f64 87.8

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

   9. Applied egg-rr 87.8%

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

   10. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}}{x}\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \frac{{z}^{2}}{x}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \frac{{z}^{2}}{x}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot \frac{{z}^{2}}{x}\right) \]

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

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \color{blue}{\frac{{z}^{2}}{x}}\right) \]

       6. unpow2 N/A

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

       7. *-lowering-*.f64 93.4

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

   12. Simplified 93.4%

       \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\left(0.0007936500793651 + y\right) \cdot \frac{z \cdot z}{x}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, x \cdot \left(\log x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x + -1, x, \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x + -1, x, \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3e-6)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma (+ (log x) -1.0) x (* (+ y 0.0007936500793651) (/ (* z z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e-6) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma((log(x) + -1.0), x, ((y + 0.0007936500793651) * ((z * z) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3e-6)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(Float64(log(x) + -1.0), x, Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3e-6], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision] * x + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0000000000000001e-6

   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 99.7

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

   5. Simplified 99.7%

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

   if 3.0000000000000001e-6 < x

   1. Initial program 88.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(0 - x\right)} + \frac{91893853320467}{100000000000000}\right)\right) \]

      20. --lowering--.f64 88.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(0 - x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. remove-double-neg N/A

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

      17. log-lowering-log.f64 88.1

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

   7. Simplified 88.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{x \cdot \left(-1 + \log x\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-lowering-+.f64 88.0

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

   9. Applied egg-rr 88.0%

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

   10. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot {z}^{2}}}{x}\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \frac{{z}^{2}}{x}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \frac{{z}^{2}}{x}}\right) \]

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

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right)} \cdot \frac{{z}^{2}}{x}\right) \]

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

          \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \left(\frac{7936500793651}{10000000000000000} + y\right) \cdot \color{blue}{\frac{{z}^{2}}{x}}\right) \]

       6. unpow2 N/A

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

       7. *-lowering-*.f64 93.5

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

   12. Simplified 93.5%

       \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\left(0.0007936500793651 + y\right) \cdot \frac{z \cdot z}{x}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x + -1, x, \left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 114:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x + -1, x, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 114.0)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma
    (+ (log x) -1.0)
    x
    (/
     (fma z (fma z 0.0007936500793651 -0.0027777777777778) 0.083333333333333)
     x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 114.0) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma((log(x) + -1.0), x, (fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 114.0)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(Float64(log(x) + -1.0), x, Float64(fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 114.0], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision] * x + N[(N[(z * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 114

   1. Initial program 99.6%

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

   3. 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 99.6

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

   5. Simplified 99.6%

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

   if 114 < x

   1. Initial program 88.0%

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

      1. +-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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(0 - x\right)} + \frac{91893853320467}{100000000000000}\right)\right) \]

      20. --lowering--.f64 88.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(0 - x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. remove-double-neg N/A

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

      17. log-lowering-log.f64 87.9

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

   7. Simplified 87.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{x \cdot \left(-1 + \log x\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-lowering-+.f64 87.9

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

   9. Applied egg-rr 87.9%

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

   10. Taylor expanded in y around 0

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

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

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

       2. +-commutative N/A

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

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

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

       4. sub-neg N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. accelerator-lowering-fma.f64 86.9

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

   12. Simplified 86.9%

       \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 114:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x + -1, x, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x + -1, x, \frac{y \cdot \left(z \cdot z\right)}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.7e+27)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (fma (+ (log x) -1.0) x (/ (* y (* z z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.7e+27) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = fma((log(x) + -1.0), x, ((y * (z * z)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.7e+27)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = fma(Float64(log(x) + -1.0), x, Float64(Float64(y * Float64(z * z)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.7e+27], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision] * x + N[(N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.69999999999999976e27

   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 98.4

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

   5. Simplified 98.4%

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

   if 4.69999999999999976e27 < x

   1. Initial program 87.3%

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

      1. +-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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(0 - x\right)} + \frac{91893853320467}{100000000000000}\right)\right) \]

      20. --lowering--.f64 87.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(0 - x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. remove-double-neg N/A

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

      17. log-lowering-log.f64 87.3

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

   7. Simplified 87.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{x \cdot \left(-1 + \log x\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-lowering-+.f64 87.3

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

   9. Applied egg-rr 87.3%

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

   10. Taylor expanded in y around inf

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 84.7

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

   12. Simplified 84.7%

       \[\leadsto \mathsf{fma}\left(-1 + \log x, x, \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x + -1, x, \frac{y \cdot \left(z \cdot z\right)}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.7e+35)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (* x (+ (log x) -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.7e+35) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = x * (log(x) + -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.7e+35)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(x * Float64(log(x) + -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.7e+35], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7000000000000001e35

   1. Initial program 99.6%

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

   3. 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 97.7

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

   5. Simplified 97.7%

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

   if 1.7000000000000001e35 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in 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. mul-1-neg N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} - 1\right) \]

      2. log-rec N/A

         \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) - 1\right) \]

      3. remove-double-neg N/A

         \[\leadsto x \cdot \left(\color{blue}{\log x} - 1\right) \]

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

         \[\leadsto \color{blue}{x \cdot \left(\log x - 1\right)} \]

      5. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\log x + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      6. metadata-eval N/A

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

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

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

      8. log-lowering-log.f64 80.9

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

   5. Simplified 80.9%

      \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          0.083333333333333
          (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))))
   (if (<= t_0 -5e+36)
     (* (* z y) (/ z x))
     (if (<= t_0 2e+82)
       (/
        (fma
         z
         (fma z 0.0007936500793651 -0.0027777777777778)
         0.083333333333333)
        x)
       (* (+ y 0.0007936500793651) (/ (* z z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
	double tmp;
	if (t_0 <= -5e+36) {
		tmp = (z * y) * (z / x);
	} else if (t_0 <= 2e+82) {
		tmp = fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = (y + 0.0007936500793651) * ((z * z) / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_0 <= -5e+36)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_0 <= 2e+82)
		tmp = Float64(fma(z, fma(z, 0.0007936500793651, -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(Float64(y + 0.0007936500793651) * Float64(Float64(z * z) / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+36], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+82], N[(N[(z * N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

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

         \[\leadsto \frac{\color{blue}{y \cdot {z}^{2}}}{x} \]

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.0

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

   5. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 73.2

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

   7. Applied egg-rr 73.2%

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

   if -4.99999999999999977e36 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1.9999999999999999e82

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 51.2

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

   8. Simplified 51.2%

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

   if 1.9999999999999999e82 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 90.3%

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

      1. +-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-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + \frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \mathsf{fma}\left(x + \frac{-1}{2}, \log x, \color{blue}{\left(0 - x\right)} + \frac{91893853320467}{100000000000000}\right)\right) \]

      20. --lowering--.f64 90.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \color{blue}{\left(0 - x\right)} + 0.91893853320467\right)\right) \]

   4. Applied egg-rr 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(x + -0.5, \log x, \left(0 - x\right) + 0.91893853320467\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. remove-double-neg N/A

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

      17. log-lowering-log.f64 90.3

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

   7. Simplified 90.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right), \frac{1}{x}, \color{blue}{x \cdot \left(-1 + \log x\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. un-div-inv N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-lowering-+.f64 90.3

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

   9. Applied egg-rr 90.3%

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

   10. Taylor expanded in z around inf

       \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   11. Step-by-step derivation

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2} + \frac{y}{x} \cdot {z}^{2}} \]

       2. associate-*l/ N/A

          \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot {z}^{2} + \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} \cdot {z}^{2} + \frac{y \cdot {z}^{2}}{x} \]

       4. metadata-eval N/A

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

       5. associate-*l/ N/A

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

       6. associate-*r/ N/A

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

       7. associate-/l* N/A

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

       8. distribute-rgt-out N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

       13. +-lowering-+.f64 73.6

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

   12. Simplified 73.6%

       \[\leadsto \color{blue}{\frac{z \cdot z}{x} \cdot \left(0.0007936500793651 + y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(y + 0.0007936500793651\right) \cdot \frac{z \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          0.083333333333333
          (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))))
   (if (<= t_0 -5e+36)
     (* (* z y) (/ z x))
     (if (<= t_0 5e+23)
       (fma (/ 1.0 x) 0.083333333333333 0.91893853320467)
       (* z (* 0.0007936500793651 (/ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
	double tmp;
	if (t_0 <= -5e+36) {
		tmp = (z * y) * (z / x);
	} else if (t_0 <= 5e+23) {
		tmp = fma((1.0 / x), 0.083333333333333, 0.91893853320467);
	} else {
		tmp = z * (0.0007936500793651 * (z / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
	tmp = 0.0
	if (t_0 <= -5e+36)
		tmp = Float64(Float64(z * y) * Float64(z / x));
	elseif (t_0 <= 5e+23)
		tmp = fma(Float64(1.0 / x), 0.083333333333333, 0.91893853320467);
	else
		tmp = Float64(z * Float64(0.0007936500793651 * Float64(z / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+36], N[(N[(z * y), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+23], N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + 0.91893853320467), $MachinePrecision], N[(z * N[(0.0007936500793651 * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 86.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

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

         \[\leadsto \frac{\color{blue}{y \cdot {z}^{2}}}{x} \]

      3. unpow2 N/A

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

      4. *-lowering-*.f64 69.0

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

   5. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z \cdot z\right)}{x}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 73.2

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

   7. Applied egg-rr 73.2%

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

   if -4.99999999999999977e36 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.9999999999999999e23

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate--l+ N/A

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

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

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 97.9

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 50.1

         \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]

   8. Simplified 50.1%

      \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} + \frac{91893853320467}{100000000000000} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} + \frac{91893853320467}{100000000000000} \]

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

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

      5. /-lowering-/.f64 50.1

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

   10. Applied egg-rr 50.1%

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

   if 4.9999999999999999e23 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

   5. Simplified 86.3%

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

   6. Taylor expanded in z around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

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

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

   8. Simplified 81.4%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000}} \]

       2. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{x}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{x}} \]

       4. metadata-eval N/A

          \[\leadsto {z}^{2} \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} \cdot 1}}{x} \]

       5. associate-*r/ N/A

          \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \]

       6. unpow2 N/A

          \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \]

       7. associate-*l* N/A

          \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)\right)} \]

       8. associate-*r/ N/A

          \[\leadsto z \cdot \left(z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \]

       9. metadata-eval N/A

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

       10. associate-*r/ N/A

           \[\leadsto z \cdot \color{blue}{\frac{z \cdot \frac{7936500793651}{10000000000000000}}{x}} \]

       11. *-commutative N/A

           \[\leadsto z \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} \cdot z}}{x} \]

       12. associate-*r/ N/A

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

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

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

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

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

       15. /-lowering-/.f64 64.8

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

   11. Simplified 64.8%

       \[\leadsto \color{blue}{z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{z}{x}\\ \mathbf{elif}\;0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)) 5e+23)
   (fma (/ 1.0 x) 0.083333333333333 0.91893853320467)
   (* z (* 0.0007936500793651 (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) <= 5e+23) {
		tmp = fma((1.0 / x), 0.083333333333333, 0.91893853320467);
	} else {
		tmp = z * (0.0007936500793651 * (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)) <= 5e+23)
		tmp = fma(Float64(1.0 / x), 0.083333333333333, 0.91893853320467);
	else
		tmp = Float64(z * Float64(0.0007936500793651 * Float64(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], 5e+23], N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + 0.91893853320467), $MachinePrecision], N[(z * N[(0.0007936500793651 * N[(z / x), $MachinePrecision]), $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) < 4.9999999999999999e23

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 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. associate--l+ N/A

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

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

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 78.5

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

   5. Simplified 78.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 37.1

         \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]

   8. Simplified 37.1%

      \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} + \frac{91893853320467}{100000000000000} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} + \frac{91893853320467}{100000000000000} \]

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

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

      5. /-lowering-/.f64 37.1

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

   10. Applied egg-rr 37.1%

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

   if 4.9999999999999999e23 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

   5. Simplified 86.3%

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

   6. Taylor expanded in z around -inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

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

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

   8. Simplified 81.4%

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

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{{z}^{2}}{x} \cdot \frac{7936500793651}{10000000000000000}} \]

       2. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{{z}^{2} \cdot \frac{7936500793651}{10000000000000000}}{x}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000}}{x}} \]

       4. metadata-eval N/A

          \[\leadsto {z}^{2} \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} \cdot 1}}{x} \]

       5. associate-*r/ N/A

          \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \]

       6. unpow2 N/A

          \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \]

       7. associate-*l* N/A

          \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)\right)} \]

       8. associate-*r/ N/A

          \[\leadsto z \cdot \left(z \cdot \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \]

       9. metadata-eval N/A

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

       10. associate-*r/ N/A

           \[\leadsto z \cdot \color{blue}{\frac{z \cdot \frac{7936500793651}{10000000000000000}}{x}} \]

       11. *-commutative N/A

           \[\leadsto z \cdot \frac{\color{blue}{\frac{7936500793651}{10000000000000000} \cdot z}}{x} \]

       12. associate-*r/ N/A

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

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

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

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

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

       15. /-lowering-/.f64 64.8

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

   11. Simplified 64.8%

       \[\leadsto \color{blue}{z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(0.0007936500793651 \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7e+103)
   (/
    (fma
     z
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     0.083333333333333)
    x)
   (* z (* z (+ (/ 0.0007936500793651 x) (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7e+103) {
		tmp = fma(z, fma(z, (y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x;
	} else {
		tmp = z * (z * ((0.0007936500793651 / x) + (y / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7e+103)
		tmp = Float64(fma(z, fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), 0.083333333333333) / x);
	else
		tmp = Float64(z * Float64(z * Float64(Float64(0.0007936500793651 / x) + Float64(y / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7e+103], N[(N[(z * N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(z * N[(z * N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e103

   1. Initial program 99.6%

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

   3. 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 89.9

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

   5. Simplified 89.9%

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

   if 7e103 < x

   1. Initial program 84.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

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

         \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

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

         \[\leadsto z \cdot \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \]

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

         \[\leadsto z \cdot \left(z \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)}\right) \]

      6. associate-*r/ N/A

         \[\leadsto z \cdot \left(z \cdot \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} + \frac{y}{x}\right)\right) \]

      7. metadata-eval N/A

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

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

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

      9. /-lowering-/.f64 17.8

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

   5. Simplified 17.8%

      \[\leadsto \color{blue}{z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.7% 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 94.2%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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 61.8

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

5. Simplified 61.8%

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

6. Final simplification 61.8%

   \[\leadsto \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), 0.083333333333333\right)}{x} \]
7. Add Preprocessing

Alternative 14: 24.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (/ 1.0 x) 0.083333333333333 0.91893853320467))

C

double code(double x, double y, double z) {
	return fma((1.0 / x), 0.083333333333333, 0.91893853320467);
}

Julia

function code(x, y, z)
	return fma(Float64(1.0 / x), 0.083333333333333, 0.91893853320467)
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333 + 0.91893853320467), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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. associate--l+ N/A

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

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

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

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

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 60.0

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

5. Simplified 60.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 25.6

      \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]

8. Simplified 25.6%

   \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x} + \frac{91893853320467}{100000000000000}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{83333333333333}{1000000000000000}}}} + \frac{91893853320467}{100000000000000} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{83333333333333}{1000000000000000}} + \frac{91893853320467}{100000000000000} \]

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

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

   5. /-lowering-/.f64 25.6

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

10. Applied egg-rr 25.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x}, 0.083333333333333, 0.91893853320467\right)} \]
11. Add Preprocessing

Alternative 15: 24.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ 0.91893853320467 + \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 0.91893853320467 (/ 0.083333333333333 x)))

C

double code(double x, double y, double z) {
	return 0.91893853320467 + (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 = 0.91893853320467d0 + (0.083333333333333d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return 0.91893853320467 + (0.083333333333333 / x);
}

Python

def code(x, y, z):
	return 0.91893853320467 + (0.083333333333333 / x)

Julia

function code(x, y, z)
	return Float64(0.91893853320467 + Float64(0.083333333333333 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.91893853320467 + (0.083333333333333 / x);
end

Wolfram

code[x_, y_, z_] := N[(0.91893853320467 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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. associate--l+ N/A

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

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

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

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

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 60.0

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

5. Simplified 60.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 25.6

      \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]

8. Simplified 25.6%

   \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]
9. Add Preprocessing

Alternative 16: 24.3% 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 94.2%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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. associate--l+ N/A

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

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

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

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

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 60.0

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

5. Simplified 60.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 25.0

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]

8. Simplified 25.0%

   \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
9. Add Preprocessing

Alternative 17: 4.0% accurate, 148.0× speedup

Math

\[\begin{array}{l} \\ 0.91893853320467 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.91893853320467)

C

double code(double x, double y, double z) {
	return 0.91893853320467;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.91893853320467d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.91893853320467;
}

Python

def code(x, y, z):
	return 0.91893853320467

Julia

function code(x, y, z)
	return 0.91893853320467
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.91893853320467;
end

Wolfram

code[x_, y_, z_] := 0.91893853320467

Derivation

1. Initial program 94.2%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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. associate--l+ N/A

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

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

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

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

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. /-lowering-/.f64 60.0

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

5. Simplified 60.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{91893853320467}{100000000000000} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 25.6

      \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]

8. Simplified 25.6%

   \[\leadsto 0.91893853320467 + \color{blue}{\frac{0.083333333333333}{x}} \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{91893853320467}{100000000000000}} \]
10. Step-by-step derivation

11. Simplified 3.8%

    \[\leadsto \color{blue}{0.91893853320467} \]
12. Add Preprocessing

Developer Target 1: 98.6% 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 2024195 
(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)))