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

Percentage Accurate: 94.3% → 99.5%

Time: 14.3s

Alternatives: 12

Speedup: 1.0×

• 29.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+47}:\\ \;\;\;\;\left(0.91893853320467 - \mathsf{fma}\left(\log x, 0.5 - x, x\right)\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.7e+47)
   (+
    (- 0.91893853320467 (fma (log x) (- 0.5 x) x))
    (/
     (fma
      z
      (fma (+ y 0.0007936500793651) z -0.0027777777777778)
      0.083333333333333)
     x))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/ (+ y 0.0007936500793651) (/ (/ x z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.7e+47) {
		tmp = (0.91893853320467 - fma(log(x), (0.5 - x), x)) + (fma(z, fma((y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.7e+47)
		tmp = Float64(Float64(0.91893853320467 - fma(log(x), Float64(0.5 - x), x)) + Float64(fma(z, fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), 0.083333333333333) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(y + 0.0007936500793651) / Float64(Float64(x / z) / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.7e+47], N[(N[(0.91893853320467 - N[(N[Log[x], $MachinePrecision] * N[(0.5 - x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6999999999999999e47

   1. Initial program 99.7%

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

      1. remove-double-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+r- 99.7%

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

      4. remove-double-neg 99.7%

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

      5. sub-neg 99.7%

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

      6. associate--r+ 99.7%

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

      7. *-commutative 99.7%

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

      8. distribute-rgt-neg-in 99.7%

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

      9. fma-def 99.8%

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

      10. neg-sub0 99.8%

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

      11. associate-+l- 99.8%

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

      12. neg-sub0 99.8%

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

      13. +-commutative 99.8%

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

      14. unsub-neg 99.8%

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

      15. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   if 1.6999999999999999e47 < 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. Taylor expanded in z around inf 87.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 87.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 88.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 88.4%

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

      4. associate-/r* 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+47}:\\ \;\;\;\;\left(0.91893853320467 - \mathsf{fma}\left(\log x, 0.5 - x, x\right)\right) + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+33}:\\ \;\;\;\;t_0 + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))))
   (if (<= x 5e+33)
     (+
      t_0
      (/
       (+
        0.083333333333333
        (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
       x))
     (+ t_0 (/ (+ y 0.0007936500793651) (/ (/ x z) z))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 5e+33) {
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	} else {
		tmp = t_0 + ((y + 0.0007936500793651) / ((x / z) / z));
	}
	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 = 0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)
    if (x <= 5d+33) then
        tmp = t_0 + ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x)
    else
        tmp = t_0 + ((y + 0.0007936500793651d0) / ((x / z) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 5e+33) {
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	} else {
		tmp = t_0 + ((y + 0.0007936500793651) / ((x / z) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)
	tmp = 0
	if x <= 5e+33:
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x)
	else:
		tmp = t_0 + ((y + 0.0007936500793651) / ((x / z) / z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x))
	tmp = 0.0
	if (x <= 5e+33)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x));
	else
		tmp = Float64(t_0 + Float64(Float64(y + 0.0007936500793651) / Float64(Float64(x / z) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	tmp = 0.0;
	if (x <= 5e+33)
		tmp = t_0 + ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x);
	else
		tmp = t_0 + ((y + 0.0007936500793651) / ((x / z) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+33], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(y + 0.0007936500793651), $MachinePrecision] / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999973e33

   1. Initial program 99.7%

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

   if 4.99999999999999973e33 < 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. Taylor expanded in z around inf 87.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 87.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 88.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 88.8%

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

      4. associate-/r* 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}}\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\log x + -1\right)\\ \mathbf{if}\;x \leq 4.1 \cdot 10^{+33}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}} + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ (log x) -1.0))))
   (if (<= x 4.1e+33)
     (+
      (/
       (+
        0.083333333333333
        (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
       x)
      t_0)
     (+ (/ (+ y 0.0007936500793651) (/ (/ x z) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (log(x) + -1.0);
	double tmp;
	if (x <= 4.1e+33) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + t_0;
	} else {
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + 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 = x * (log(x) + (-1.0d0))
    if (x <= 4.1d+33) then
        tmp = ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x) + t_0
    else
        tmp = ((y + 0.0007936500793651d0) / ((x / z) / z)) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (Math.log(x) + -1.0);
	double tmp;
	if (x <= 4.1e+33) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + t_0;
	} else {
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (math.log(x) + -1.0)
	tmp = 0
	if x <= 4.1e+33:
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + t_0
	else:
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(log(x) + -1.0))
	tmp = 0.0
	if (x <= 4.1e+33)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x) + t_0);
	else
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) / Float64(Float64(x / z) / z)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (log(x) + -1.0);
	tmp = 0.0;
	if (x <= 4.1e+33)
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + t_0;
	else
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.1e+33], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.09999999999999995e33

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.0%

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

      1. *-commutative 36.4%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]

      2. sub-neg 36.4%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      3. metadata-eval 36.4%

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

      4. mul-1-neg 36.4%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + -1\right) + \frac{0.083333333333333}{x} \]

      5. log-rec 36.4%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + -1\right) + \frac{0.083333333333333}{x} \]

      6. remove-double-neg 36.4%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + -1\right) + \frac{0.083333333333333}{x} \]

   4. Simplified 98.0%

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

   if 4.09999999999999995e33 < 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. Taylor expanded in z around inf 87.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 87.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 88.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 88.8%

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

      4. associate-/r* 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 73.8%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]

      2. sub-neg 73.8%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      3. metadata-eval 73.8%

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

      4. mul-1-neg 73.8%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + -1\right) + \frac{0.083333333333333}{x} \]

      5. log-rec 73.8%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + -1\right) + \frac{0.083333333333333}{x} \]

      6. remove-double-neg 73.8%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + -1\right) + \frac{0.083333333333333}{x} \]

   7. Simplified 99.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+33}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}} + x \cdot \left(\log x + -1\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}} + x \cdot \left(\log x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.1e-6)
   (+
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
     x)
    (+ 0.91893853320467 (* (log x) -0.5)))
   (+ (/ (+ y 0.0007936500793651) (/ (/ x z) z)) (* x (+ (log x) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.1e-6) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (log(x) * -0.5));
	} else {
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + (x * (log(x) + -1.0));
	}
	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) :: tmp
    if (x <= 4.1d-6) then
        tmp = ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x) + (0.91893853320467d0 + (log(x) * (-0.5d0)))
    else
        tmp = ((y + 0.0007936500793651d0) / ((x / z) / z)) + (x * (log(x) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.1e-6) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (Math.log(x) * -0.5));
	} else {
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + (x * (Math.log(x) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.1e-6:
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (math.log(x) * -0.5))
	else:
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + (x * (math.log(x) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.1e-6)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x) + Float64(0.91893853320467 + Float64(log(x) * -0.5)));
	else
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) / Float64(Float64(x / z) / z)) + Float64(x * Float64(log(x) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.1e-6)
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (log(x) * -0.5));
	else
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + (x * (log(x) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.1e-6], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0999999999999997e-6

   1. Initial program 99.7%

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

      1. add-cbrt-cube 99.7%

         \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) \cdot \left(\left(x - 0.5\right) \cdot \log x - x\right)}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. pow3 99.7%

         \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(\left(x - 0.5\right) \cdot \log x - x\right)}^{3}}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. fma-neg 99.7%

         \[\leadsto \left(\sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right)\right)}}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. sub-neg 99.7%

         \[\leadsto \left(\sqrt[3]{{\left(\mathsf{fma}\left(\color{blue}{x + \left(-0.5\right)}, \log x, -x\right)\right)}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(\sqrt[3]{{\left(\mathsf{fma}\left(x + \color{blue}{-0.5}, \log x, -x\right)\right)}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x + -0.5, \log x, -x\right)\right)}^{3}}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Taylor expanded in x around 0 99.7%

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

   if 4.0999999999999997e-6 < x

   1. Initial program 89.5%

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

   2. Taylor expanded in z around inf 89.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 89.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 89.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 89.7%

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

      4. associate-/r* 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in x around inf 98.0%

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

      1. *-commutative 67.9%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]

      2. sub-neg 67.9%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      3. metadata-eval 67.9%

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

      4. mul-1-neg 67.9%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + -1\right) + \frac{0.083333333333333}{x} \]

      5. log-rec 67.9%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + -1\right) + \frac{0.083333333333333}{x} \]

      6. remove-double-neg 67.9%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + -1\right) + \frac{0.083333333333333}{x} \]

   7. Simplified 98.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}} + x \cdot \left(\log x + -1\right)\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{y + 0.0007936500793651}{\frac{x}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.1e-6)
   (+
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
     x)
    (+ 0.91893853320467 (* (log x) -0.5)))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (/ (+ y 0.0007936500793651) (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.1e-6) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (log(x) * -0.5));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((y + 0.0007936500793651) / (x / z)));
	}
	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) :: tmp
    if (x <= 4.1d-6) then
        tmp = ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x) + (0.91893853320467d0 + (log(x) * (-0.5d0)))
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * ((y + 0.0007936500793651d0) / (x / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.1e-6) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (Math.log(x) * -0.5));
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * ((y + 0.0007936500793651) / (x / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.1e-6:
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (math.log(x) * -0.5))
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * ((y + 0.0007936500793651) / (x / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.1e-6)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x) + Float64(0.91893853320467 + Float64(log(x) * -0.5)));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(Float64(y + 0.0007936500793651) / Float64(x / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.1e-6)
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (log(x) * -0.5));
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * ((y + 0.0007936500793651) / (x / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.1e-6], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(y + 0.0007936500793651), $MachinePrecision] / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0999999999999997e-6

   1. Initial program 99.7%

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

      1. add-cbrt-cube 99.7%

         \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) \cdot \left(\left(x - 0.5\right) \cdot \log x - x\right)}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. pow3 99.7%

         \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(\left(x - 0.5\right) \cdot \log x - x\right)}^{3}}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. fma-neg 99.7%

         \[\leadsto \left(\sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right)\right)}}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. sub-neg 99.7%

         \[\leadsto \left(\sqrt[3]{{\left(\mathsf{fma}\left(\color{blue}{x + \left(-0.5\right)}, \log x, -x\right)\right)}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(\sqrt[3]{{\left(\mathsf{fma}\left(x + \color{blue}{-0.5}, \log x, -x\right)\right)}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x + -0.5, \log x, -x\right)\right)}^{3}}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Taylor expanded in x around 0 99.7%

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

   if 4.0999999999999997e-6 < x

   1. Initial program 89.5%

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

   2. Taylor expanded in z around inf 89.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 89.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 89.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 89.7%

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

      4. associate-/r* 99.0%

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

   4. Simplified 99.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{\frac{x}{z}}{z}}} \]
   5. Step-by-step derivation

      1. associate-/r/ 99.0%

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

      2. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{y + 0.0007936500793651}{\frac{x}{z}}\\ \end{array} \]

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.1e-6)
   (+
    (/
     (+
      0.083333333333333
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
     x)
    (+ 0.91893853320467 (* (log x) -0.5)))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/ (+ y 0.0007936500793651) (/ (/ x z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.1e-6) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (log(x) * -0.5));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z));
	}
	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) :: tmp
    if (x <= 4.1d-6) then
        tmp = ((0.083333333333333d0 + (z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))) / x) + (0.91893853320467d0 + (log(x) * (-0.5d0)))
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + ((y + 0.0007936500793651d0) / ((x / z) / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.1e-6) {
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (Math.log(x) * -0.5));
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.1e-6:
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (math.log(x) * -0.5))
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.1e-6)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))) / x) + Float64(0.91893853320467 + Float64(log(x) * -0.5)));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(Float64(y + 0.0007936500793651) / Float64(Float64(x / z) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.1e-6)
		tmp = ((0.083333333333333 + (z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))) / x) + (0.91893853320467 + (log(x) * -0.5));
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + ((y + 0.0007936500793651) / ((x / z) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.1e-6], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0999999999999997e-6

   1. Initial program 99.7%

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

      1. add-cbrt-cube 99.7%

         \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x - x\right)\right) \cdot \left(\left(x - 0.5\right) \cdot \log x - x\right)}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      2. pow3 99.7%

         \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(\left(x - 0.5\right) \cdot \log x - x\right)}^{3}}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      3. fma-neg 99.7%

         \[\leadsto \left(\sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right)\right)}}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      4. sub-neg 99.7%

         \[\leadsto \left(\sqrt[3]{{\left(\mathsf{fma}\left(\color{blue}{x + \left(-0.5\right)}, \log x, -x\right)\right)}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(\sqrt[3]{{\left(\mathsf{fma}\left(x + \color{blue}{-0.5}, \log x, -x\right)\right)}^{3}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x + -0.5, \log x, -x\right)\right)}^{3}}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Taylor expanded in x around 0 99.7%

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

   if 4.0999999999999997e-6 < x

   1. Initial program 89.5%

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

   2. Taylor expanded in z around inf 89.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 89.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 89.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 89.7%

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

      4. associate-/r* 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + \log x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}}\\ \end{array} \]

Alternative 7: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-95} \lor \neg \left(z \leq 6 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.55e-95) (not (<= z 6e-28)))
   (+ (/ (+ y 0.0007936500793651) (/ (/ x z) z)) (* x (+ (log x) -1.0)))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e-95) || !(z <= 6e-28)) {
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + (x * (log(x) + -1.0));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 / x);
	}
	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) :: tmp
    if ((z <= (-1.55d-95)) .or. (.not. (z <= 6d-28))) then
        tmp = ((y + 0.0007936500793651d0) / ((x / z) / z)) + (x * (log(x) + (-1.0d0)))
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e-95) || !(z <= 6e-28)) {
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + (x * (Math.log(x) + -1.0));
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.55e-95) or not (z <= 6e-28):
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + (x * (math.log(x) + -1.0))
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.55e-95) || !(z <= 6e-28))
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) / Float64(Float64(x / z) / z)) + Float64(x * Float64(log(x) + -1.0)));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.55e-95) || ~((z <= 6e-28)))
		tmp = ((y + 0.0007936500793651) / ((x / z) / z)) + (x * (log(x) + -1.0));
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.55e-95], N[Not[LessEqual[z, 6e-28]], $MachinePrecision]], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.54999999999999996e-95 or 6.00000000000000005e-28 < z

   1. Initial program 91.3%

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

   2. Taylor expanded in z around inf 89.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 89.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 90.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 90.4%

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

      4. associate-/r* 98.2%

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

   4. Simplified 98.2%

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

   5. Taylor expanded in x around inf 97.8%

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

      1. *-commutative 29.5%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]

      2. sub-neg 29.5%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      3. metadata-eval 29.5%

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

      4. mul-1-neg 29.5%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + -1\right) + \frac{0.083333333333333}{x} \]

      5. log-rec 29.5%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + -1\right) + \frac{0.083333333333333}{x} \]

      6. remove-double-neg 29.5%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + -1\right) + \frac{0.083333333333333}{x} \]

   7. Simplified 97.8%

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

   if -1.54999999999999996e-95 < z < 6.00000000000000005e-28

   1. Initial program 99.5%

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

   2. Taylor expanded in z around 0 97.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-95} \lor \neg \left(z \leq 6 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{y + 0.0007936500793651}{\frac{\frac{x}{z}}{z}} + x \cdot \left(\log x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 8: 70.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+78} \lor \neg \left(z \leq 5.8 \cdot 10^{+19}\right):\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \frac{z \cdot \left(z \cdot 0.0007936500793651\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4e+78) (not (<= z 5.8e+19)))
   (+
    (+ 0.91893853320467 (* (log x) -0.5))
    (/ (* z (* z 0.0007936500793651)) x))
   (+ (* x (+ (log x) -1.0)) (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4e+78) || !(z <= 5.8e+19)) {
		tmp = (0.91893853320467 + (log(x) * -0.5)) + ((z * (z * 0.0007936500793651)) / x);
	} else {
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	}
	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) :: tmp
    if ((z <= (-4d+78)) .or. (.not. (z <= 5.8d+19))) then
        tmp = (0.91893853320467d0 + (log(x) * (-0.5d0))) + ((z * (z * 0.0007936500793651d0)) / x)
    else
        tmp = (x * (log(x) + (-1.0d0))) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4e+78) || !(z <= 5.8e+19)) {
		tmp = (0.91893853320467 + (Math.log(x) * -0.5)) + ((z * (z * 0.0007936500793651)) / x);
	} else {
		tmp = (x * (Math.log(x) + -1.0)) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4e+78) or not (z <= 5.8e+19):
		tmp = (0.91893853320467 + (math.log(x) * -0.5)) + ((z * (z * 0.0007936500793651)) / x)
	else:
		tmp = (x * (math.log(x) + -1.0)) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4e+78) || !(z <= 5.8e+19))
		tmp = Float64(Float64(0.91893853320467 + Float64(log(x) * -0.5)) + Float64(Float64(z * Float64(z * 0.0007936500793651)) / x));
	else
		tmp = Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4e+78) || ~((z <= 5.8e+19)))
		tmp = (0.91893853320467 + (log(x) * -0.5)) + ((z * (z * 0.0007936500793651)) / x);
	else
		tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4e+78], N[Not[LessEqual[z, 5.8e+19]], $MachinePrecision]], N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(z * 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.00000000000000003e78 or 5.8e19 < z

   1. Initial program 88.6%

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

   2. Taylor expanded in z around inf 88.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 88.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 89.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 89.4%

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

      4. associate-/r* 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 61.7%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{0.0007936500793651 \cdot \frac{{z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. *-commutative 61.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2}}{x} \cdot 0.0007936500793651} \]

      2. associate-*l/ 61.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot 0.0007936500793651}{x}} \]

      3. unpow2 61.7%

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

      4. associate-*l* 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around 0 51.8%

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

   if -4.00000000000000003e78 < z < 5.8e19

   1. Initial program 99.5%

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

   2. Taylor expanded in z around 0 84.8%

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

   3. Taylor expanded in x around inf 83.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} + \frac{0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. *-commutative 83.0%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]

      2. sub-neg 83.0%

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

      3. metadata-eval 83.0%

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

      4. mul-1-neg 83.0%

         \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + -1\right) + \frac{0.083333333333333}{x} \]

      5. log-rec 83.0%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + -1\right) + \frac{0.083333333333333}{x} \]

      6. remove-double-neg 83.0%

         \[\leadsto x \cdot \left(\color{blue}{\log x} + -1\right) + \frac{0.083333333333333}{x} \]

   5. Simplified 83.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+78} \lor \neg \left(z \leq 5.8 \cdot 10^{+19}\right):\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \frac{z \cdot \left(z \cdot 0.0007936500793651\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 9: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+77} \lor \neg \left(z \leq 3.55 \cdot 10^{+19}\right):\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \frac{z \cdot \left(z \cdot 0.0007936500793651\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.08e+77) (not (<= z 3.55e+19)))
   (+
    (+ 0.91893853320467 (* (log x) -0.5))
    (/ (* z (* z 0.0007936500793651)) x))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.08e+77) || !(z <= 3.55e+19)) {
		tmp = (0.91893853320467 + (log(x) * -0.5)) + ((z * (z * 0.0007936500793651)) / x);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 / x);
	}
	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) :: tmp
    if ((z <= (-1.08d+77)) .or. (.not. (z <= 3.55d+19))) then
        tmp = (0.91893853320467d0 + (log(x) * (-0.5d0))) + ((z * (z * 0.0007936500793651d0)) / x)
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.08e+77) || !(z <= 3.55e+19)) {
		tmp = (0.91893853320467 + (Math.log(x) * -0.5)) + ((z * (z * 0.0007936500793651)) / x);
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.08e+77) or not (z <= 3.55e+19):
		tmp = (0.91893853320467 + (math.log(x) * -0.5)) + ((z * (z * 0.0007936500793651)) / x)
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.08e+77) || !(z <= 3.55e+19))
		tmp = Float64(Float64(0.91893853320467 + Float64(log(x) * -0.5)) + Float64(Float64(z * Float64(z * 0.0007936500793651)) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.08e+77) || ~((z <= 3.55e+19)))
		tmp = (0.91893853320467 + (log(x) * -0.5)) + ((z * (z * 0.0007936500793651)) / x);
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.08e+77], N[Not[LessEqual[z, 3.55e+19]], $MachinePrecision]], N[(N[(0.91893853320467 + N[(N[Log[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(z * 0.0007936500793651), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.07999999999999996e77 or 3.55e19 < z

   1. Initial program 88.6%

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

   2. Taylor expanded in z around inf 88.6%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}} \]
   3. Step-by-step derivation

      1. *-commutative 88.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(0.0007936500793651 + y\right) \cdot {z}^{2}}}{x} \]

      2. associate-/l* 89.4%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{0.0007936500793651 + y}{\frac{x}{{z}^{2}}}} \]

      3. unpow2 89.4%

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

      4. associate-/r* 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in y around 0 61.7%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{0.0007936500793651 \cdot \frac{{z}^{2}}{x}} \]
   6. Step-by-step derivation

      1. *-commutative 61.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2}}{x} \cdot 0.0007936500793651} \]

      2. associate-*l/ 61.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{{z}^{2} \cdot 0.0007936500793651}{x}} \]

      3. unpow2 61.7%

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

      4. associate-*l* 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around 0 51.8%

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

   if -1.07999999999999996e77 < z < 3.55e19

   1. Initial program 99.5%

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

   2. Taylor expanded in z around 0 84.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{+77} \lor \neg \left(z \leq 3.55 \cdot 10^{+19}\right):\\ \;\;\;\;\left(0.91893853320467 + \log x \cdot -0.5\right) + \frac{z \cdot \left(z \cdot 0.0007936500793651\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 10: 34.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\log x + -1\right) + \frac{-0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (* x (+ (log x) -1.0)) (/ -0.083333333333333 x)))

C

double code(double x, double y, double z) {
	return (x * (log(x) + -1.0)) + (-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 * (log(x) + (-1.0d0))) + ((-0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.log(x) + -1.0)) + (-0.083333333333333 / x);
}

Python

def code(x, y, z):
	return (x * (math.log(x) + -1.0)) + (-0.083333333333333 / x)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(-0.083333333333333 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (log(x) + -1.0)) + (-0.083333333333333 / x);
end

Wolfram

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

Derivation

1. Initial program 94.3%

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

2. Taylor expanded in z around 0 54.6%

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

3. Taylor expanded in x around inf 53.6%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} + \frac{0.083333333333333}{x} \]
4. Step-by-step derivation

   1. *-commutative 53.6%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]

   2. sub-neg 53.6%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

   3. metadata-eval 53.6%

      \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   4. mul-1-neg 53.6%

      \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + -1\right) + \frac{0.083333333333333}{x} \]

   5. log-rec 53.6%

      \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + -1\right) + \frac{0.083333333333333}{x} \]

   6. remove-double-neg 53.6%

      \[\leadsto x \cdot \left(\color{blue}{\log x} + -1\right) + \frac{0.083333333333333}{x} \]

5. Simplified 53.6%

   \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]
6. Step-by-step derivation

   1. div-inv 53.6%

      \[\leadsto x \cdot \left(\log x + -1\right) + \color{blue}{0.083333333333333 \cdot \frac{1}{x}} \]

   2. frac-2neg 53.6%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \color{blue}{\frac{-1}{-x}} \]

   3. metadata-eval 53.6%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{\color{blue}{-1}}{-x} \]

   4. add-sqr-sqrt 0.0%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \]

   5. sqrt-unprod 40.0%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} \]

   6. sqr-neg 40.0%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\sqrt{\color{blue}{x \cdot x}}} \]

   7. sqrt-unprod 36.7%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

   8. add-sqr-sqrt 36.7%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\color{blue}{x}} \]

7. Applied egg-rr 36.7%

   \[\leadsto x \cdot \left(\log x + -1\right) + \color{blue}{0.083333333333333 \cdot \frac{-1}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 36.7%

      \[\leadsto x \cdot \left(\log x + -1\right) + \color{blue}{\frac{0.083333333333333 \cdot -1}{x}} \]

   2. metadata-eval 36.7%

      \[\leadsto x \cdot \left(\log x + -1\right) + \frac{\color{blue}{-0.083333333333333}}{x} \]

9. Simplified 36.7%

   \[\leadsto x \cdot \left(\log x + -1\right) + \color{blue}{\frac{-0.083333333333333}{x}} \]

10. Final simplification 36.7%

    \[\leadsto x \cdot \left(\log x + -1\right) + \frac{-0.083333333333333}{x} \]

Alternative 11: 55.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (* x (+ (log x) -1.0)) (/ 0.083333333333333 x)))

C

double code(double x, double y, double z) {
	return (x * (log(x) + -1.0)) + (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 * (log(x) + (-1.0d0))) + (0.083333333333333d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.log(x) + -1.0)) + (0.083333333333333 / x);
}

Python

def code(x, y, z):
	return (x * (math.log(x) + -1.0)) + (0.083333333333333 / x)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(log(x) + -1.0)) + Float64(0.083333333333333 / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (log(x) + -1.0)) + (0.083333333333333 / x);
end

Wolfram

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

Derivation

1. Initial program 94.3%

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

2. Taylor expanded in z around 0 54.6%

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

3. Taylor expanded in x around inf 53.6%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} + \frac{0.083333333333333}{x} \]
4. Step-by-step derivation

   1. *-commutative 53.6%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]

   2. sub-neg 53.6%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

   3. metadata-eval 53.6%

      \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   4. mul-1-neg 53.6%

      \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + -1\right) + \frac{0.083333333333333}{x} \]

   5. log-rec 53.6%

      \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + -1\right) + \frac{0.083333333333333}{x} \]

   6. remove-double-neg 53.6%

      \[\leadsto x \cdot \left(\color{blue}{\log x} + -1\right) + \frac{0.083333333333333}{x} \]

5. Simplified 53.6%

   \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]

6. Final simplification 53.6%

   \[\leadsto x \cdot \left(\log x + -1\right) + \frac{0.083333333333333}{x} \]

Alternative 12: 1.5% accurate, 41.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ -0.083333333333333 x))

C

double code(double x, double y, double z) {
	return -0.083333333333333 / x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-0.083333333333333d0) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

2. Taylor expanded in z around 0 54.6%

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

3. Taylor expanded in x around inf 53.6%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} + \frac{0.083333333333333}{x} \]
4. Step-by-step derivation

   1. *-commutative 53.6%

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} + \frac{0.083333333333333}{x} \]

   2. sub-neg 53.6%

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(-1\right)\right)} + \frac{0.083333333333333}{x} \]

   3. metadata-eval 53.6%

      \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{-1}\right) + \frac{0.083333333333333}{x} \]

   4. mul-1-neg 53.6%

      \[\leadsto x \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + -1\right) + \frac{0.083333333333333}{x} \]

   5. log-rec 53.6%

      \[\leadsto x \cdot \left(\left(-\color{blue}{\left(-\log x\right)}\right) + -1\right) + \frac{0.083333333333333}{x} \]

   6. remove-double-neg 53.6%

      \[\leadsto x \cdot \left(\color{blue}{\log x} + -1\right) + \frac{0.083333333333333}{x} \]

5. Simplified 53.6%

   \[\leadsto \color{blue}{x \cdot \left(\log x + -1\right)} + \frac{0.083333333333333}{x} \]
6. Step-by-step derivation

   1. div-inv 53.6%

      \[\leadsto x \cdot \left(\log x + -1\right) + \color{blue}{0.083333333333333 \cdot \frac{1}{x}} \]

   2. frac-2neg 53.6%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \color{blue}{\frac{-1}{-x}} \]

   3. metadata-eval 53.6%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{\color{blue}{-1}}{-x} \]

   4. add-sqr-sqrt 0.0%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \]

   5. sqrt-unprod 40.0%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} \]

   6. sqr-neg 40.0%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\sqrt{\color{blue}{x \cdot x}}} \]

   7. sqrt-unprod 36.7%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

   8. add-sqr-sqrt 36.7%

      \[\leadsto x \cdot \left(\log x + -1\right) + 0.083333333333333 \cdot \frac{-1}{\color{blue}{x}} \]

7. Applied egg-rr 36.7%

   \[\leadsto x \cdot \left(\log x + -1\right) + \color{blue}{0.083333333333333 \cdot \frac{-1}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 36.7%

      \[\leadsto x \cdot \left(\log x + -1\right) + \color{blue}{\frac{0.083333333333333 \cdot -1}{x}} \]

   2. metadata-eval 36.7%

      \[\leadsto x \cdot \left(\log x + -1\right) + \frac{\color{blue}{-0.083333333333333}}{x} \]

9. Simplified 36.7%

   \[\leadsto x \cdot \left(\log x + -1\right) + \color{blue}{\frac{-0.083333333333333}{x}} \]

10. Taylor expanded in x around 0 1.5%

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

11. Final simplification 1.5%

    \[\leadsto \frac{-0.083333333333333}{x} \]

Developer target: 98.6% accurate, 1.0× 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 2023230 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))