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

Percentage Accurate: 93.8% → 99.4%

Time: 14.0s

Alternatives: 18

Speedup: 1.0×

• 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: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{x}, \left(\log x - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2e-8)
   (fma
    (fma
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     z
     0.083333333333333)
    (/ 1.0 x)
    (* (- (log x) 1.0) x))
   (-
    (fma
     (fma (/ y x) z (/ (fma 0.0007936500793651 z -0.0027777777777778) x))
     z
     (fma (- x 0.5) (log x) (/ 0.083333333333333 x)))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2e-8) {
		tmp = fma(fma(fma(z, (y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), (1.0 / x), ((log(x) - 1.0) * x));
	} else {
		tmp = fma(fma((y / x), z, (fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma((x - 0.5), log(x), (0.083333333333333 / x))) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2e-8)
		tmp = fma(fma(fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), Float64(1.0 / x), Float64(Float64(log(x) - 1.0) * x));
	else
		tmp = Float64(fma(fma(Float64(y / x), z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma(Float64(x - 0.5), log(x), Float64(0.083333333333333 / x))) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2e-8], N[(N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] * z + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-8

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.7

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-log.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 2e-8 < 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. Taylor expanded in z around 0

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{x}, \left(\log x - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ t_1 := \left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(0.083333333333333, \frac{1}{x}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
        (t_1 (* (* (/ z x) z) (+ y 0.0007936500793651))))
   (if (<= t_0 -5e+85)
     t_1
     (if (<= t_0 1e+308)
       (fma
        0.083333333333333
        (/ 1.0 x)
        (fma (log x) (- x 0.5) (- 0.91893853320467 x)))
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	double t_1 = ((z / x) * z) * (y + 0.0007936500793651);
	double tmp;
	if (t_0 <= -5e+85) {
		tmp = t_1;
	} else if (t_0 <= 1e+308) {
		tmp = fma(0.083333333333333, (1.0 / x), fma(log(x), (x - 0.5), (0.91893853320467 - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467))
	t_1 = Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651))
	tmp = 0.0
	if (t_0 <= -5e+85)
		tmp = t_1;
	elseif (t_0 <= 1e+308)
		tmp = fma(0.083333333333333, Float64(1.0 / x), fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+85], t$95$1, If[LessEqual[t$95$0, 1e+308], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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)) < -5.0000000000000001e85 or 1e308 < (+.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 83.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   9. 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-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      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. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if -5.0000000000000001e85 < (+.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)) < 1e308

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.4

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 88.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{0.083333333333333}, \frac{1}{x}, \mathsf{fma}\left(\log x, x - 0.5, \left(-x\right) + 0.91893853320467\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{elif}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(0.083333333333333, \frac{1}{x}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ t_1 := \left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
        (t_1 (* (* (/ z x) z) (+ y 0.0007936500793651))))
   (if (<= t_0 -5e+85)
     t_1
     (if (<= t_0 1e+308)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 0.083333333333333 x) 0.91893853320467) x))
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	double t_1 = ((z / x) * z) * (y + 0.0007936500793651);
	double tmp;
	if (t_0 <= -5e+85) {
		tmp = t_1;
	} else if (t_0 <= 1e+308) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467))
	t_1 = Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651))
	tmp = 0.0
	if (t_0 <= -5e+85)
		tmp = t_1;
	elseif (t_0 <= 1e+308)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+85], t$95$1, If[LessEqual[t$95$0, 1e+308], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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)) < -5.0000000000000001e85 or 1e308 < (+.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 83.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   9. 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-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      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. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if -5.0000000000000001e85 < (+.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)) < 1e308

   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-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{elif}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ t_1 := \left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) - \left(x - 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
        (t_1 (* (* (/ z x) z) (+ y 0.0007936500793651))))
   (if (<= t_0 -5e+85)
     t_1
     (if (<= t_0 1e+308)
       (-
        (fma (log x) (+ -0.5 x) (/ 0.083333333333333 x))
        (- x 0.91893853320467))
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	double t_1 = ((z / x) * z) * (y + 0.0007936500793651);
	double tmp;
	if (t_0 <= -5e+85) {
		tmp = t_1;
	} else if (t_0 <= 1e+308) {
		tmp = fma(log(x), (-0.5 + x), (0.083333333333333 / x)) - (x - 0.91893853320467);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467))
	t_1 = Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651))
	tmp = 0.0
	if (t_0 <= -5e+85)
		tmp = t_1;
	elseif (t_0 <= 1e+308)
		tmp = Float64(fma(log(x), Float64(-0.5 + x), Float64(0.083333333333333 / x)) - Float64(x - 0.91893853320467));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+85], t$95$1, If[LessEqual[t$95$0, 1e+308], N[(N[(N[Log[x], $MachinePrecision] * N[(-0.5 + x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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)) < -5.0000000000000001e85 or 1e308 < (+.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 83.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   9. 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-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      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. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if -5.0000000000000001e85 < (+.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)) < 1e308

   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 inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 7.6

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

   8. Applied rewrites 88.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{elif}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(\log x, -0.5 + x, \frac{0.083333333333333}{x}\right) - \left(x - 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ t_1 := \left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(0.083333333333333, \frac{1}{x}, \left(\log x - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
        (t_1 (* (* (/ z x) z) (+ y 0.0007936500793651))))
   (if (<= t_0 -5e+85)
     t_1
     (if (<= t_0 1e+308)
       (fma 0.083333333333333 (/ 1.0 x) (* (- (log x) 1.0) x))
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	double t_1 = ((z / x) * z) * (y + 0.0007936500793651);
	double tmp;
	if (t_0 <= -5e+85) {
		tmp = t_1;
	} else if (t_0 <= 1e+308) {
		tmp = fma(0.083333333333333, (1.0 / x), ((log(x) - 1.0) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467))
	t_1 = Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651))
	tmp = 0.0
	if (t_0 <= -5e+85)
		tmp = t_1;
	elseif (t_0 <= 1e+308)
		tmp = fma(0.083333333333333, Float64(1.0 / x), Float64(Float64(log(x) - 1.0) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+85], t$95$1, If[LessEqual[t$95$0, 1e+308], N[(0.083333333333333 * N[(1.0 / x), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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)) < -5.0000000000000001e85 or 1e308 < (+.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 83.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   9. 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-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      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. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if -5.0000000000000001e85 < (+.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)) < 1e308

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.4

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-log.f64 97.9

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{83333333333333}{1000000000000000}}, \frac{1}{x}, \left(\log x - 1\right) \cdot x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 86.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \mathbf{elif}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(0.083333333333333, \frac{1}{x}, \left(\log x - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{x}, \left(\log x - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + 0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 4e+287)
   (fma
    (fma
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     z
     0.083333333333333)
    (/ 1.0 x)
    (* (- (log x) 1.0) x))
   (* (* (/ (+ y 0.0007936500793651) x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 4e+287) {
		tmp = fma(fma(fma(z, (y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), (1.0 / x), ((log(x) - 1.0) * x));
	} else {
		tmp = (((y + 0.0007936500793651) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 4e+287)
		tmp = fma(fma(fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), Float64(1.0 / x), Float64(Float64(log(x) - 1.0) * x));
	else
		tmp = Float64(Float64(Float64(Float64(y + 0.0007936500793651) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 4e+287], N[(N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $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) < 4.0000000000000003e287

   1. Initial program 97.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 97.6

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-log.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if 4.0000000000000003e287 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 77.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 78.1

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

   5. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(z \cdot z\right) \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 87.2%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 4 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{x}, \left(\log x - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + 0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 950000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + 0.0007936500793651}{x} \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 950000000000.0)
   (fma
    (fma
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     z
     0.083333333333333)
    (/ 1.0 x)
    (fma (log x) (- x 0.5) (- 0.91893853320467 x)))
   (+
    (* (* (/ (+ y 0.0007936500793651) x) z) z)
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 950000000000.0) {
		tmp = fma(fma(fma(z, (y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), (1.0 / x), fma(log(x), (x - 0.5), (0.91893853320467 - x)));
	} else {
		tmp = ((((y + 0.0007936500793651) / x) * z) * z) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 950000000000.0)
		tmp = fma(fma(fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), Float64(1.0 / x), fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) / x) * z) * z) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 950000000000.0], N[(N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5e11

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.6

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if 9.5e11 < x

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   7. 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-/l* 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. lower-*.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. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.6

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 950000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{x}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + 0.0007936500793651}{x} \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 900000000000:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + 0.0007936500793651}{x} \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 900000000000.0)
   (fma
    (- x 0.5)
    (log x)
    (-
     (+
      (/
       (fma
        (fma z (+ y 0.0007936500793651) -0.0027777777777778)
        z
        0.083333333333333)
       x)
      0.91893853320467)
     x))
   (+
    (* (* (/ (+ y 0.0007936500793651) x) z) z)
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 900000000000.0) {
		tmp = fma((x - 0.5), log(x), (((fma(fma(z, (y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333) / x) + 0.91893853320467) - x));
	} else {
		tmp = ((((y + 0.0007936500793651) / x) * z) * z) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 900000000000.0)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(fma(fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333) / x) + 0.91893853320467) - x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) / x) * z) * z) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 900000000000.0], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9e11

   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. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-+l+ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 9e11 < x

   1. Initial program 87.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   7. 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-/l* 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. lower-*.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. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.6

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 900000000000:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + 0.0007936500793651}{x} \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{x}, \left(\log x - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + 0.0007936500793651}{x} \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.029)
   (fma
    (fma
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     z
     0.083333333333333)
    (/ 1.0 x)
    (* (- (log x) 1.0) x))
   (+
    (* (* (/ (+ y 0.0007936500793651) x) z) z)
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.029) {
		tmp = fma(fma(fma(z, (y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), (1.0 / x), ((log(x) - 1.0) * x));
	} else {
		tmp = ((((y + 0.0007936500793651) / x) * z) * z) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.029)
		tmp = fma(fma(fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), Float64(1.0 / x), Float64(Float64(log(x) - 1.0) * x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) / x) * z) * z) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.029], N[(N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.7

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

      19. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. lower-log.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if 0.0290000000000000015 < x

   1. Initial program 87.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \color{blue}{\frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}} \]
   7. 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-/l* 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. lower-*.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. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.3

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

   8. Applied rewrites 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{x}, \left(\log x - 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + 0.0007936500793651}{x} \cdot z\right) \cdot z + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1250:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{y}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1250.0)
   (/
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+ (* (* z z) (/ y x)) (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1250.0) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z * z) * (y / x)) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1250.0)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z * z) * Float64(y / x)) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1250.0], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1250

   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. lower-/.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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 1250 < x

   1. Initial program 87.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 84.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1250:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{y}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \left(z \cdot z\right) \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 14500000000000.0)
   (/
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (+ (* (- (log x) 1.0) x) (* (* z z) (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 14500000000000.0) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((log(x) - 1.0) * x) + ((z * z) * (y / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 14500000000000.0)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(log(x) - 1.0) * x) + Float64(Float64(z * z) * Float64(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 14500000000000.0], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e13

   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. lower-/.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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if 1.45e13 < 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. Taylor expanded in z around inf

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 85.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{y}{x} \cdot \left(z \cdot z\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. mul-1-neg N/A

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

       3. log-rec N/A

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

       4. remove-double-neg N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-log.f64 85.4

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

   11. Applied rewrites 85.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 14500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x + \left(z \cdot z\right) \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.8e+44)
   (/
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.8e+44) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.8e+44)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.8e+44], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.8e44

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if 6.8e44 < x

   1. Initial program 87.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 13: 64.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{+85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 2e+85)
   (/
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ z x) z) (+ y 0.0007936500793651))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e+85) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * z) * (y + 0.0007936500793651);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e+85)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 2e+85], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $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) < 2e85

   1. Initial program 97.0%

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

   3. 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. lower-/.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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 53.5

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

   5. Applied rewrites 53.5%

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

   if 2e85 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 86.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   9. 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-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      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. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 74.7

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

   10. Applied rewrites 74.7%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{+85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{if}\;y + 0.0007936500793651 \leq -2000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.001:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ z x) z) y)))
   (if (<= (+ y 0.0007936500793651) -2000.0)
     t_0
     (if (<= (+ y 0.0007936500793651) 0.001)
       (* (* (/ 0.0007936500793651 x) z) z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z / x) * z) * y;
	double tmp;
	if ((y + 0.0007936500793651) <= -2000.0) {
		tmp = t_0;
	} else if ((y + 0.0007936500793651) <= 0.001) {
		tmp = ((0.0007936500793651 / x) * z) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((z / x) * z) * y
    if ((y + 0.0007936500793651d0) <= (-2000.0d0)) then
        tmp = t_0
    else if ((y + 0.0007936500793651d0) <= 0.001d0) then
        tmp = ((0.0007936500793651d0 / x) * z) * z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((z / x) * z) * y;
	double tmp;
	if ((y + 0.0007936500793651) <= -2000.0) {
		tmp = t_0;
	} else if ((y + 0.0007936500793651) <= 0.001) {
		tmp = ((0.0007936500793651 / x) * z) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((z / x) * z) * y
	tmp = 0
	if (y + 0.0007936500793651) <= -2000.0:
		tmp = t_0
	elif (y + 0.0007936500793651) <= 0.001:
		tmp = ((0.0007936500793651 / x) * z) * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z / x) * z) * y)
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -2000.0)
		tmp = t_0;
	elseif (Float64(y + 0.0007936500793651) <= 0.001)
		tmp = Float64(Float64(Float64(0.0007936500793651 / x) * z) * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((z / x) * z) * y;
	tmp = 0.0;
	if ((y + 0.0007936500793651) <= -2000.0)
		tmp = t_0;
	elseif ((y + 0.0007936500793651) <= 0.001)
		tmp = ((0.0007936500793651 / x) * z) * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -2000.0], t$95$0, If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 0.001], N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -2e3 or 1e-3 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 90.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 46.6

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 47.1%

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

   9. Applied rewrites 47.6%

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

   if -2e3 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 1e-3

   1. Initial program 95.3%

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

   3. Taylor expanded in z around 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. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 37.9%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -2000:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.001:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* (/ z x) z) (+ y 0.0007936500793651)))

C

double code(double x, double y, double z) {
	return ((z / x) * z) * (y + 0.0007936500793651);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((z / x) * z) * (y + 0.0007936500793651d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((z / x) * z) * (y + 0.0007936500793651);
}

Python

def code(x, y, z):
	return ((z / x) * z) * (y + 0.0007936500793651)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z / x) * z) * (y + 0.0007936500793651);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 29.7

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

5. Applied rewrites 29.7%

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

7. Applied rewrites 31.1%

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

8. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
9. 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-*r/ N/A

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

   3. metadata-eval N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. associate-*l/ N/A

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

   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. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 42.8

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

10. Applied rewrites 42.8%

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

11. Final simplification 42.8%

    \[\leadsto \left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right) \]
12. Add Preprocessing

Alternative 16: 41.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(y + 0.0007936500793651\right) \cdot z\right) \cdot z}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (* (+ y 0.0007936500793651) z) z) x))

C

double code(double x, double y, double z) {
	return (((y + 0.0007936500793651) * z) * z) / 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 = (((y + 0.0007936500793651d0) * z) * z) / x
end function

Java

public static double code(double x, double y, double z) {
	return (((y + 0.0007936500793651) * z) * z) / x;
}

Python

def code(x, y, z):
	return (((y + 0.0007936500793651) * z) * z) / x

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) * z) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((y + 0.0007936500793651) * z) * z) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 93.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 39.3

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

5. Applied rewrites 39.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 40.6%

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

Alternative 17: 26.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.0007936500793651}{x} \cdot z\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* (/ 0.0007936500793651 x) z) z))

C

double code(double x, double y, double z) {
	return ((0.0007936500793651 / x) * z) * z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((0.0007936500793651d0 / x) * z) * z
end function

Java

public static double code(double x, double y, double z) {
	return ((0.0007936500793651 / x) * z) * z;
}

Python

def code(x, y, z):
	return ((0.0007936500793651 / x) * z) * z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(0.0007936500793651 / x) * z) * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((0.0007936500793651 / x) * z) * z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 93.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 39.3

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

5. Applied rewrites 39.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 26.1%

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

Alternative 18: 26.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* (/ z x) 0.0007936500793651) z))

C

double code(double x, double y, double z) {
	return ((z / x) * 0.0007936500793651) * z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((z / x) * 0.0007936500793651d0) * z
end function

Java

public static double code(double x, double y, double z) {
	return ((z / x) * 0.0007936500793651) * z;
}

Python

def code(x, y, z):
	return ((z / x) * 0.0007936500793651) * z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z / x) * 0.0007936500793651) * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z / x) * 0.0007936500793651) * z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z / x), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 93.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
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. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 39.3

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

5. Applied rewrites 39.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 26.1%

   \[\leadsto \left(\frac{0.0007936500793651}{x} \cdot z\right) \cdot \color{blue}{z} \]
9. Step-by-step derivation

10. Applied rewrites 26.1%

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

11. Final simplification 26.1%

    \[\leadsto \left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z \]
12. Add Preprocessing

Developer Target 1: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024235 
(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)))