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

Percentage Accurate: 93.7% → 99.5%

Time: 15.3s

Alternatives: 12

Speedup: 1.0×

• 27.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.7% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -0.5\right) \cdot \log x\\ \mathbf{if}\;x \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\left(\frac{{t_0}^{3} - {x}^{3}}{{t_0}^{2} + x \cdot \left(x + t_0\right)} + 0.91893853320467\right) + \frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x -0.5) (log x))))
   (if (<= x 5e+43)
     (+
      (+
       (/ (- (pow t_0 3.0) (pow x 3.0)) (+ (pow t_0 2.0) (* x (+ x t_0))))
       0.91893853320467)
      (/
       (+
        (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
        0.083333333333333)
       x))
     (+
      (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
      (* z (* z (+ (/ y x) (/ 0.0007936500793651 x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + -0.5) * log(x);
	double tmp;
	if (x <= 5e+43) {
		tmp = (((pow(t_0, 3.0) - pow(x, 3.0)) / (pow(t_0, 2.0) + (x * (x + t_0)))) + 0.91893853320467) + (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / 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) :: t_0
    real(8) :: tmp
    t_0 = (x + (-0.5d0)) * log(x)
    if (x <= 5d+43) then
        tmp = ((((t_0 ** 3.0d0) - (x ** 3.0d0)) / ((t_0 ** 2.0d0) + (x * (x + t_0)))) + 0.91893853320467d0) + (((z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)) + 0.083333333333333d0) / x)
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * (z * ((y / x) + (0.0007936500793651d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + -0.5) * Math.log(x);
	double tmp;
	if (x <= 5e+43) {
		tmp = (((Math.pow(t_0, 3.0) - Math.pow(x, 3.0)) / (Math.pow(t_0, 2.0) + (x * (x + t_0)))) + 0.91893853320467) + (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x);
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + -0.5) * math.log(x)
	tmp = 0
	if x <= 5e+43:
		tmp = (((math.pow(t_0, 3.0) - math.pow(x, 3.0)) / (math.pow(t_0, 2.0) + (x * (x + t_0)))) + 0.91893853320467) + (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x)
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + -0.5) * log(x))
	tmp = 0.0
	if (x <= 5e+43)
		tmp = Float64(Float64(Float64(Float64((t_0 ^ 3.0) - (x ^ 3.0)) / Float64((t_0 ^ 2.0) + Float64(x * Float64(x + t_0)))) + 0.91893853320467) + Float64(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + -0.5) * log(x);
	tmp = 0.0;
	if (x <= 5e+43)
		tmp = ((((t_0 ^ 3.0) - (x ^ 3.0)) / ((t_0 ^ 2.0) + (x * (x + t_0)))) + 0.91893853320467) + (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x);
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+43], N[(N[(N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(x * N[(x + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $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[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000004e43

   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. flip3-- 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. pow2 99.7%

         \[\leadsto \left(\frac{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{3} - {x}^{3}}{\color{blue}{{\left(\left(x - 0.5\right) \cdot \log x\right)}^{2}} + \left(x \cdot x + \left(\left(x - 0.5\right) \cdot \log x\right) \cdot x\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      5. sub-neg 99.7%

         \[\leadsto \left(\frac{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{3} - {x}^{3}}{{\left(\color{blue}{\left(x + \left(-0.5\right)\right)} \cdot \log x\right)}^{2} + \left(x \cdot x + \left(\left(x - 0.5\right) \cdot \log x\right) \cdot x\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      6. metadata-eval 99.7%

         \[\leadsto \left(\frac{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{3} - {x}^{3}}{{\left(\left(x + \color{blue}{-0.5}\right) \cdot \log x\right)}^{2} + \left(x \cdot x + \left(\left(x - 0.5\right) \cdot \log x\right) \cdot x\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      7. distribute-rgt-out 99.7%

         \[\leadsto \left(\frac{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{3} - {x}^{3}}{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{2} + \color{blue}{x \cdot \left(x + \left(x - 0.5\right) \cdot \log x\right)}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      8. sub-neg 99.7%

         \[\leadsto \left(\frac{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{3} - {x}^{3}}{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{2} + x \cdot \left(x + \color{blue}{\left(x + \left(-0.5\right)\right)} \cdot \log x\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

      9. metadata-eval 99.7%

         \[\leadsto \left(\frac{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{3} - {x}^{3}}{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{2} + x \cdot \left(x + \left(x + \color{blue}{-0.5}\right) \cdot \log x\right)} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \left(\color{blue}{\frac{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{3} - {x}^{3}}{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{2} + x \cdot \left(x + \left(x + -0.5\right) \cdot \log x\right)}} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 5.0000000000000004e43 < 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 87.5%

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

      1. +-commutative 87.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(-0.0027777777777778 \cdot \frac{z}{x} + \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      2. fma-def 87.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      3. associate-/l* 90.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}}\right) \]

      4. +-commutative 90.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2}}{\frac{x}{\color{blue}{y + 0.0007936500793651}}}\right) \]

      5. associate-/r/ 90.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{x} \cdot \left(y + 0.0007936500793651\right)}\right) \]

      6. unpow2 90.8%

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

      7. +-commutative 90.8%

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

   4. Simplified 90.8%

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

   5. Taylor expanded in z around inf 90.7%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 90.7%

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

      2. *-commutative 90.7%

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

      3. associate-*l* 99.5%

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

      4. associate-*r/ 99.5%

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

      5. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\left(\frac{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{3} - {x}^{3}}{{\left(\left(x + -0.5\right) \cdot \log x\right)}^{2} + x \cdot \left(x + \left(x + -0.5\right) \cdot \log x\right)} + 0.91893853320467\right) + \frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 2: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right)\\ t_1 := 0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+63}:\\ \;\;\;\;t_1 + y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;t_0 \leq 2000000:\\ \;\;\;\;t_1 + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t_1 + z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))
        (t_1 (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))))
   (if (<= t_0 -1e+63)
     (+ t_1 (* y (* z (/ z x))))
     (if (<= t_0 2000000.0)
       (+ t_1 (/ 0.083333333333333 x))
       (+ t_1 (* z (* z (/ 0.0007936500793651 x))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	double tmp;
	if (t_0 <= -1e+63) {
		tmp = t_1 + (y * (z * (z / x)));
	} else if (t_0 <= 2000000.0) {
		tmp = t_1 + (0.083333333333333 / x);
	} else {
		tmp = t_1 + (z * (z * (0.0007936500793651 / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)
    t_1 = 0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)
    if (t_0 <= (-1d+63)) then
        tmp = t_1 + (y * (z * (z / x)))
    else if (t_0 <= 2000000.0d0) then
        tmp = t_1 + (0.083333333333333d0 / x)
    else
        tmp = t_1 + (z * (z * (0.0007936500793651d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	double t_1 = 0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x);
	double tmp;
	if (t_0 <= -1e+63) {
		tmp = t_1 + (y * (z * (z / x)));
	} else if (t_0 <= 2000000.0) {
		tmp = t_1 + (0.083333333333333 / x);
	} else {
		tmp = t_1 + (z * (z * (0.0007936500793651 / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)
	t_1 = 0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)
	tmp = 0
	if t_0 <= -1e+63:
		tmp = t_1 + (y * (z * (z / x)))
	elif t_0 <= 2000000.0:
		tmp = t_1 + (0.083333333333333 / x)
	else:
		tmp = t_1 + (z * (z * (0.0007936500793651 / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778))
	t_1 = Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x))
	tmp = 0.0
	if (t_0 <= -1e+63)
		tmp = Float64(t_1 + Float64(y * Float64(z * Float64(z / x))));
	elseif (t_0 <= 2000000.0)
		tmp = Float64(t_1 + Float64(0.083333333333333 / x));
	else
		tmp = Float64(t_1 + Float64(z * Float64(z * Float64(0.0007936500793651 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778);
	t_1 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	tmp = 0.0;
	if (t_0 <= -1e+63)
		tmp = t_1 + (y * (z * (z / x)));
	elseif (t_0 <= 2000000.0)
		tmp = t_1 + (0.083333333333333 / x);
	else
		tmp = t_1 + (z * (z * (0.0007936500793651 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+63], N[(t$95$1 + N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2000000.0], N[(t$95$1 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(z * N[(z * N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) < -1.00000000000000006e63

   1. Initial program 90.6%

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

   2. Taylor expanded in y around inf 90.0%

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

      1. associate-/l* 96.8%

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

      2. unpow2 96.8%

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

   4. Simplified 96.8%

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

      1. div-inv 96.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{y \cdot \frac{1}{\frac{x}{z \cdot z}}} \]

      2. clear-num 96.9%

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

      3. associate-/l* 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-/r/ 99.2%

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

   8. Simplified 99.2%

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

   if -1.00000000000000006e63 < (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z) < 2e6

   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 98.5%

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

   if 2e6 < (*.f64 (-.f64 (*.f64 (+.f64 y 7936500793651/10000000000000000) z) 13888888888889/5000000000000000) z)

   1. Initial program 88.7%

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

   2. Taylor expanded in z around inf 76.6%

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

      1. +-commutative 76.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(-0.0027777777777778 \cdot \frac{z}{x} + \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      2. fma-def 76.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      3. associate-/l* 77.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}}\right) \]

      4. +-commutative 77.6%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2}}{\frac{x}{\color{blue}{y + 0.0007936500793651}}}\right) \]

      5. associate-/r/ 77.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{x} \cdot \left(y + 0.0007936500793651\right)}\right) \]

      6. unpow2 77.7%

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

      7. +-commutative 77.7%

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

   4. Simplified 77.7%

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

   5. Taylor expanded in z around inf 89.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*l* 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around 0 93.8%

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

      1. associate-*r/ 93.8%

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

      2. associate-*l/ 93.8%

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

      3. *-commutative 93.8%

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

   10. Simplified 93.8%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq -1 \cdot 10^{+63}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \leq 2000000:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \frac{0.0007936500793651}{x}\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ \mathbf{if}\;x \leq 300000:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))))
   (if (<= x 300000.0)
     (+
      (/
       (+
        (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
        0.083333333333333)
       x)
      t_0)
     (+ t_0 (* z (* z (+ (/ y x) (/ 0.0007936500793651 x))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	double tmp;
	if (x <= 300000.0) {
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	} else {
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / 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) :: t_0
    real(8) :: tmp
    t_0 = 0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)
    if (x <= 300000.0d0) then
        tmp = (((z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + t_0
    else
        tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651d0 / x))))
    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 <= 300000.0) {
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	} else {
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)
	tmp = 0
	if x <= 300000.0:
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0
	else:
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	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 <= 300000.0)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0);
	else
		tmp = Float64(t_0 + Float64(z * Float64(z * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)))));
	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 <= 300000.0)
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + t_0;
	else
		tmp = t_0 + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	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, 300000.0], N[(N[(N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[(z * N[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3e5

   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 3e5 < x

   1. Initial program 89.2%

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

   2. Taylor expanded in z around inf 89.2%

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

      1. +-commutative 89.2%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(-0.0027777777777778 \cdot \frac{z}{x} + \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      2. fma-def 89.2%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      3. associate-/l* 92.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}}\right) \]

      4. +-commutative 92.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2}}{\frac{x}{\color{blue}{y + 0.0007936500793651}}}\right) \]

      5. associate-/r/ 92.0%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{x} \cdot \left(y + 0.0007936500793651\right)}\right) \]

      6. unpow2 92.0%

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

      7. +-commutative 92.0%

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

   4. Simplified 92.0%

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

   5. Taylor expanded in z around inf 92.0%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*l* 99.5%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 300000:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 4: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ t_1 := 0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{+108}:\\ \;\;\;\;t_1 + y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{+30} \lor \neg \left(z \leq 5.1 \cdot 10^{+20}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ 0.91893853320467 (* -0.5 (log x)))
          (/ (* z z) (/ x (+ y 0.0007936500793651)))))
        (t_1 (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))))
   (if (<= z -5.2e+154)
     t_0
     (if (<= z -2.25e+108)
       (+ t_1 (* y (* z (/ z x))))
       (if (or (<= z -8.8e+30) (not (<= z 5.1e+20)))
         t_0
         (+ t_1 (/ 0.083333333333333 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + (-0.5 * log(x))) + ((z * z) / (x / (y + 0.0007936500793651)));
	double t_1 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	double tmp;
	if (z <= -5.2e+154) {
		tmp = t_0;
	} else if (z <= -2.25e+108) {
		tmp = t_1 + (y * (z * (z / x)));
	} else if ((z <= -8.8e+30) || !(z <= 5.1e+20)) {
		tmp = t_0;
	} else {
		tmp = t_1 + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.91893853320467d0 + ((-0.5d0) * log(x))) + ((z * z) / (x / (y + 0.0007936500793651d0)))
    t_1 = 0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)
    if (z <= (-5.2d+154)) then
        tmp = t_0
    else if (z <= (-2.25d+108)) then
        tmp = t_1 + (y * (z * (z / x)))
    else if ((z <= (-8.8d+30)) .or. (.not. (z <= 5.1d+20))) then
        tmp = t_0
    else
        tmp = t_1 + (0.083333333333333d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + (-0.5 * Math.log(x))) + ((z * z) / (x / (y + 0.0007936500793651)));
	double t_1 = 0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x);
	double tmp;
	if (z <= -5.2e+154) {
		tmp = t_0;
	} else if (z <= -2.25e+108) {
		tmp = t_1 + (y * (z * (z / x)));
	} else if ((z <= -8.8e+30) || !(z <= 5.1e+20)) {
		tmp = t_0;
	} else {
		tmp = t_1 + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.91893853320467 + (-0.5 * math.log(x))) + ((z * z) / (x / (y + 0.0007936500793651)))
	t_1 = 0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)
	tmp = 0
	if z <= -5.2e+154:
		tmp = t_0
	elif z <= -2.25e+108:
		tmp = t_1 + (y * (z * (z / x)))
	elif (z <= -8.8e+30) or not (z <= 5.1e+20):
		tmp = t_0
	else:
		tmp = t_1 + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(-0.5 * log(x))) + Float64(Float64(z * z) / Float64(x / Float64(y + 0.0007936500793651))))
	t_1 = Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x))
	tmp = 0.0
	if (z <= -5.2e+154)
		tmp = t_0;
	elseif (z <= -2.25e+108)
		tmp = Float64(t_1 + Float64(y * Float64(z * Float64(z / x))));
	elseif ((z <= -8.8e+30) || !(z <= 5.1e+20))
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(0.083333333333333 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.91893853320467 + (-0.5 * log(x))) + ((z * z) / (x / (y + 0.0007936500793651)));
	t_1 = 0.91893853320467 + ((log(x) * (x - 0.5)) - x);
	tmp = 0.0;
	if (z <= -5.2e+154)
		tmp = t_0;
	elseif (z <= -2.25e+108)
		tmp = t_1 + (y * (z * (z / x)));
	elseif ((z <= -8.8e+30) || ~((z <= 5.1e+20)))
		tmp = t_0;
	else
		tmp = t_1 + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(x / N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e+154], t$95$0, If[LessEqual[z, -2.25e+108], N[(t$95$1 + N[(y * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -8.8e+30], N[Not[LessEqual[z, 5.1e+20]], $MachinePrecision]], t$95$0, N[(t$95$1 + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.19999999999999978e154 or -2.25e108 < z < -8.7999999999999999e30 or 5.1e20 < z

   1. Initial program 88.5%

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

   2. Taylor expanded in z around inf 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. associate-/l* 89.4%

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

      2. unpow2 89.4%

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

   4. Simplified 89.4%

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

   5. Taylor expanded in x around 0 82.5%

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

   if -5.19999999999999978e154 < z < -2.25e108

   1. Initial program 79.0%

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

   2. Taylor expanded in y around inf 71.5%

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

      1. associate-/l* 92.5%

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

      2. unpow2 92.5%

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

   4. Simplified 92.5%

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

      1. div-inv 92.5%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{y \cdot \frac{1}{\frac{x}{z \cdot z}}} \]

      2. clear-num 92.5%

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

      3. associate-/l* 92.5%

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

   6. Applied egg-rr 92.5%

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

      1. associate-/r/ 92.5%

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

   8. Simplified 92.5%

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

   if -8.7999999999999999e30 < z < 5.1e20

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 92.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+154}:\\ \;\;\;\;\left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{+108}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + y \cdot \left(z \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{+30} \lor \neg \left(z \leq 5.1 \cdot 10^{+20}\right):\\ \;\;\;\;\left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 5: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+176}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.4e+176)
   (+
    (/
     (+
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
      0.083333333333333)
     x)
    (+ 0.91893853320467 (* x (+ (log x) -1.0))))
   (+ (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)) (* z (/ z (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.4e+176) {
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + (x * (log(x) + -1.0)));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z / (x / y)));
	}
	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 <= 2.4d+176) then
        tmp = (((z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + (0.91893853320467d0 + (x * (log(x) + (-1.0d0))))
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * (z / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.4e+176) {
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + (x * (Math.log(x) + -1.0)));
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * (z / (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 2.4e+176:
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + (x * (math.log(x) + -1.0)))
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * (z / (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.4e+176)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(x * Float64(log(x) + -1.0))));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.4e+176)
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + (x * (log(x) + -1.0)));
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.4e+176], N[(N[(N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $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[(z / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4000000000000001e176

   1. Initial program 98.2%

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

   2. Taylor expanded in x around inf 96.5%

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   3. Step-by-step derivation

      1. *-commutative 51.0%

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

      2. sub-neg 51.0%

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

      3. mul-1-neg 51.0%

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

      4. log-rec 51.0%

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

      5. remove-double-neg 51.0%

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

      6. metadata-eval 51.0%

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

   4. Simplified 96.5%

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

   if 2.4000000000000001e176 < x

   1. Initial program 81.9%

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

   2. Taylor expanded in z around inf 81.9%

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

      1. +-commutative 81.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(-0.0027777777777778 \cdot \frac{z}{x} + \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      2. fma-def 81.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      3. associate-/l* 87.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}}\right) \]

      4. +-commutative 87.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2}}{\frac{x}{\color{blue}{y + 0.0007936500793651}}}\right) \]

      5. associate-/r/ 87.9%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{x} \cdot \left(y + 0.0007936500793651\right)}\right) \]

      6. unpow2 87.9%

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

      7. +-commutative 87.9%

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

   4. Simplified 87.9%

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

   5. Taylor expanded in z around inf 87.9%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-*l* 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 90.9%

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

      1. *-commutative 90.9%

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

      2. associate-/l* 93.7%

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

   10. Simplified 93.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+176}:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \frac{z}{\frac{x}{y}}\\ \end{array} \]

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0075:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + -0.5 \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0075)
   (+
    (/
     (+
      (* z (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))
      0.083333333333333)
     x)
    (+ 0.91893853320467 (* -0.5 (log x))))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (* z (* z (+ (/ y x) (/ 0.0007936500793651 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0075) {
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + (-0.5 * log(x)));
	} else {
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / 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 (x <= 0.0075d0) then
        tmp = (((z * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0)) + 0.083333333333333d0) / x) + (0.91893853320467d0 + ((-0.5d0) * log(x)))
    else
        tmp = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (z * (z * ((y / x) + (0.0007936500793651d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0075) {
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + (-0.5 * Math.log(x)));
	} else {
		tmp = (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.0075:
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + (-0.5 * math.log(x)))
	else:
		tmp = (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.0075)
		tmp = Float64(Float64(Float64(Float64(z * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + Float64(0.91893853320467 + Float64(-0.5 * log(x))));
	else
		tmp = Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(z * Float64(z * Float64(Float64(y / x) + Float64(0.0007936500793651 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.0075)
		tmp = (((z * ((z * (y + 0.0007936500793651)) - 0.0027777777777778)) + 0.083333333333333) / x) + (0.91893853320467 + (-0.5 * log(x)));
	else
		tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (z * (z * ((y / x) + (0.0007936500793651 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.0075], N[(N[(N[(N[(z * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(-0.5 * N[Log[x], $MachinePrecision]), $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[(z * N[(N[(y / x), $MachinePrecision] + N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0074999999999999997

   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 0 99.1%

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

   if 0.0074999999999999997 < x

   1. Initial program 89.7%

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

   2. Taylor expanded in z around inf 89.1%

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

      1. +-commutative 89.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(-0.0027777777777778 \cdot \frac{z}{x} + \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      2. fma-def 89.1%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2} \cdot \left(0.0007936500793651 + y\right)}{x}\right)} \]

      3. associate-/l* 91.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{\frac{x}{0.0007936500793651 + y}}}\right) \]

      4. +-commutative 91.7%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \frac{{z}^{2}}{\frac{x}{\color{blue}{y + 0.0007936500793651}}}\right) \]

      5. associate-/r/ 91.8%

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \mathsf{fma}\left(-0.0027777777777778, \frac{z}{x}, \color{blue}{\frac{{z}^{2}}{x} \cdot \left(y + 0.0007936500793651\right)}\right) \]

      6. unpow2 91.8%

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

      7. +-commutative 91.8%

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

   4. Simplified 91.8%

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

   5. Taylor expanded in z around inf 91.7%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\left(\frac{y}{x} + 0.0007936500793651 \cdot \frac{1}{x}\right) \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*l* 98.9%

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

      4. associate-*r/ 98.9%

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

      5. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0075:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) + 0.083333333333333}{x} + \left(0.91893853320467 + -0.5 \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + z \cdot \left(z \cdot \left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right)\right)\\ \end{array} \]

Alternative 7: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+28} \lor \neg \left(z \leq 5.4 \cdot 10^{+21}\right):\\ \;\;\;\;\left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.85e+28) (not (<= z 5.4e+21)))
   (+
    (+ 0.91893853320467 (* -0.5 (log x)))
    (/ (* z z) (/ x (+ y 0.0007936500793651))))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.85e+28) || !(z <= 5.4e+21)) {
		tmp = (0.91893853320467 + (-0.5 * log(x))) + ((z * z) / (x / (y + 0.0007936500793651)));
	} 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 <= (-2.85d+28)) .or. (.not. (z <= 5.4d+21))) then
        tmp = (0.91893853320467d0 + ((-0.5d0) * log(x))) + ((z * z) / (x / (y + 0.0007936500793651d0)))
    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 <= -2.85e+28) || !(z <= 5.4e+21)) {
		tmp = (0.91893853320467 + (-0.5 * Math.log(x))) + ((z * z) / (x / (y + 0.0007936500793651)));
	} 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 <= -2.85e+28) or not (z <= 5.4e+21):
		tmp = (0.91893853320467 + (-0.5 * math.log(x))) + ((z * z) / (x / (y + 0.0007936500793651)))
	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 <= -2.85e+28) || !(z <= 5.4e+21))
		tmp = Float64(Float64(0.91893853320467 + Float64(-0.5 * log(x))) + Float64(Float64(z * z) / Float64(x / Float64(y + 0.0007936500793651))));
	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 <= -2.85e+28) || ~((z <= 5.4e+21)))
		tmp = (0.91893853320467 + (-0.5 * log(x))) + ((z * z) / (x / (y + 0.0007936500793651)));
	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, -2.85e+28], N[Not[LessEqual[z, 5.4e+21]], $MachinePrecision]], N[(N[(0.91893853320467 + N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(x / N[(y + 0.0007936500793651), $MachinePrecision]), $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 < -2.8500000000000001e28 or 5.4e21 < z

   1. Initial program 87.3%

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

   2. Taylor expanded in z around inf 87.4%

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

      1. associate-/l* 90.8%

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

      2. unpow2 90.8%

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

   4. Simplified 90.8%

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

   5. Taylor expanded in x around 0 78.8%

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

   if -2.8500000000000001e28 < z < 5.4e21

   1. Initial program 99.4%

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

   2. Taylor expanded in z around 0 92.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+28} \lor \neg \left(z \leq 5.4 \cdot 10^{+21}\right):\\ \;\;\;\;\left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{z \cdot z}{\frac{x}{y + 0.0007936500793651}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 8: 71.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+133} \lor \neg \left(z \leq 6.6 \cdot 10^{+23}\right):\\ \;\;\;\;\left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.3e+133) (not (<= z 6.6e+23)))
   (+ (+ 0.91893853320467 (* -0.5 (log x))) (/ y (/ x (* z z))))
   (+ (+ 0.91893853320467 (* x (+ (log x) -1.0))) (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e+133) || !(z <= 6.6e+23)) {
		tmp = (0.91893853320467 + (-0.5 * log(x))) + (y / (x / (z * z)));
	} else {
		tmp = (0.91893853320467 + (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 <= (-1.3d+133)) .or. (.not. (z <= 6.6d+23))) then
        tmp = (0.91893853320467d0 + ((-0.5d0) * log(x))) + (y / (x / (z * z)))
    else
        tmp = (0.91893853320467d0 + (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 <= -1.3e+133) || !(z <= 6.6e+23)) {
		tmp = (0.91893853320467 + (-0.5 * Math.log(x))) + (y / (x / (z * z)));
	} else {
		tmp = (0.91893853320467 + (x * (Math.log(x) + -1.0))) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.3e+133) or not (z <= 6.6e+23):
		tmp = (0.91893853320467 + (-0.5 * math.log(x))) + (y / (x / (z * z)))
	else:
		tmp = (0.91893853320467 + (x * (math.log(x) + -1.0))) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.3e+133) || !(z <= 6.6e+23))
		tmp = Float64(Float64(0.91893853320467 + Float64(-0.5 * log(x))) + Float64(y / Float64(x / Float64(z * z))));
	else
		tmp = Float64(Float64(0.91893853320467 + 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 <= -1.3e+133) || ~((z <= 6.6e+23)))
		tmp = (0.91893853320467 + (-0.5 * log(x))) + (y / (x / (z * z)));
	else
		tmp = (0.91893853320467 + (x * (log(x) + -1.0))) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.3e+133], N[Not[LessEqual[z, 6.6e+23]], $MachinePrecision]], N[(N[(0.91893853320467 + N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.91893853320467 + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2999999999999999e133 or 6.60000000000000059e23 < z

   1. Initial program 86.7%

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

   2. Taylor expanded in y around inf 61.9%

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

      1. associate-/l* 66.0%

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

      2. unpow2 66.0%

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

   4. Simplified 66.0%

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

   5. Taylor expanded in x around 0 60.3%

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

   6. Taylor expanded in x around 0 60.4%

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

   if -1.2999999999999999e133 < z < 6.60000000000000059e23

   1. Initial program 98.3%

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

   2. Taylor expanded in z around 0 86.1%

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

   3. Taylor expanded in x around inf 84.2%

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. *-commutative 84.2%

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

      2. sub-neg 84.2%

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

      3. mul-1-neg 84.2%

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

      4. log-rec 84.2%

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

      5. remove-double-neg 84.2%

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

      6. metadata-eval 84.2%

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

   5. Simplified 84.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+133} \lor \neg \left(z \leq 6.6 \cdot 10^{+23}\right):\\ \;\;\;\;\left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 9: 72.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+132} \lor \neg \left(z \leq 7.3 \cdot 10^{+22}\right):\\ \;\;\;\;\left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.2e+132) (not (<= z 7.3e+22)))
   (+ (+ 0.91893853320467 (* -0.5 (log x))) (/ y (/ x (* z z))))
   (+
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
    (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.2e+132) || !(z <= 7.3e+22)) {
		tmp = (0.91893853320467 + (-0.5 * log(x))) + (y / (x / (z * z)));
	} 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 <= (-7.2d+132)) .or. (.not. (z <= 7.3d+22))) then
        tmp = (0.91893853320467d0 + ((-0.5d0) * log(x))) + (y / (x / (z * z)))
    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 <= -7.2e+132) || !(z <= 7.3e+22)) {
		tmp = (0.91893853320467 + (-0.5 * Math.log(x))) + (y / (x / (z * z)));
	} 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 <= -7.2e+132) or not (z <= 7.3e+22):
		tmp = (0.91893853320467 + (-0.5 * math.log(x))) + (y / (x / (z * z)))
	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 <= -7.2e+132) || !(z <= 7.3e+22))
		tmp = Float64(Float64(0.91893853320467 + Float64(-0.5 * log(x))) + Float64(y / Float64(x / Float64(z * z))));
	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 <= -7.2e+132) || ~((z <= 7.3e+22)))
		tmp = (0.91893853320467 + (-0.5 * log(x))) + (y / (x / (z * z)));
	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, -7.2e+132], N[Not[LessEqual[z, 7.3e+22]], $MachinePrecision]], N[(N[(0.91893853320467 + N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x / N[(z * z), $MachinePrecision]), $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 < -7.20000000000000031e132 or 7.29999999999999979e22 < z

   1. Initial program 86.7%

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

   2. Taylor expanded in y around inf 61.9%

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

      1. associate-/l* 66.0%

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

      2. unpow2 66.0%

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

   4. Simplified 66.0%

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

   5. Taylor expanded in x around 0 60.3%

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

   6. Taylor expanded in x around 0 60.4%

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

   if -7.20000000000000031e132 < z < 7.29999999999999979e22

   1. Initial program 98.3%

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

   2. Taylor expanded in z around 0 86.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+132} \lor \neg \left(z \leq 7.3 \cdot 10^{+22}\right):\\ \;\;\;\;\left(0.91893853320467 + -0.5 \cdot \log x\right) + \frac{y}{\frac{x}{z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 10: 71.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+132} \lor \neg \left(z \leq 4.1 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}} - x\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7.5e+132) (not (<= z 4.1e+23)))
   (- (/ y (/ x (* z z))) x)
   (+ (+ 0.91893853320467 (* x (+ (log x) -1.0))) (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7.5e+132) || !(z <= 4.1e+23)) {
		tmp = (y / (x / (z * z))) - x;
	} else {
		tmp = (0.91893853320467 + (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 <= (-7.5d+132)) .or. (.not. (z <= 4.1d+23))) then
        tmp = (y / (x / (z * z))) - x
    else
        tmp = (0.91893853320467d0 + (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 <= -7.5e+132) || !(z <= 4.1e+23)) {
		tmp = (y / (x / (z * z))) - x;
	} else {
		tmp = (0.91893853320467 + (x * (Math.log(x) + -1.0))) + (0.083333333333333 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7.5e+132) or not (z <= 4.1e+23):
		tmp = (y / (x / (z * z))) - x
	else:
		tmp = (0.91893853320467 + (x * (math.log(x) + -1.0))) + (0.083333333333333 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7.5e+132) || !(z <= 4.1e+23))
		tmp = Float64(Float64(y / Float64(x / Float64(z * z))) - x);
	else
		tmp = Float64(Float64(0.91893853320467 + 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 <= -7.5e+132) || ~((z <= 4.1e+23)))
		tmp = (y / (x / (z * z))) - x;
	else
		tmp = (0.91893853320467 + (x * (log(x) + -1.0))) + (0.083333333333333 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7.5e+132], N[Not[LessEqual[z, 4.1e+23]], $MachinePrecision]], N[(N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(N[(0.91893853320467 + N[(x * N[(N[Log[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000017e132 or 4.09999999999999996e23 < z

   1. Initial program 86.7%

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

   2. Taylor expanded in y around inf 61.9%

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

      1. associate-/l* 66.0%

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

      2. unpow2 66.0%

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

   4. Simplified 66.0%

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

   5. Taylor expanded in x around 0 60.3%

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

   6. Taylor expanded in x around inf 60.3%

      \[\leadsto \color{blue}{-1 \cdot x} + \frac{y}{\frac{x}{z \cdot z}} \]
   7. Step-by-step derivation

      1. mul-1-neg 60.3%

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

   8. Simplified 60.3%

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

   if -7.50000000000000017e132 < z < 4.09999999999999996e23

   1. Initial program 98.3%

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

   2. Taylor expanded in z around 0 86.1%

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

   3. Taylor expanded in x around inf 84.2%

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \cdot x} + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]
   4. Step-by-step derivation

      1. *-commutative 84.2%

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

      2. sub-neg 84.2%

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

      3. mul-1-neg 84.2%

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

      4. log-rec 84.2%

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

      5. remove-double-neg 84.2%

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

      6. metadata-eval 84.2%

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

   5. Simplified 84.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+132} \lor \neg \left(z \leq 4.1 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}} - x\\ \mathbf{else}:\\ \;\;\;\;\left(0.91893853320467 + x \cdot \left(\log x + -1\right)\right) + \frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 11: 49.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-18} \lor \neg \left(z \leq 4.8 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.85e-18) (not (<= z 4.8e-19)))
   (- (/ y (/ x (* z z))) x)
   (/ 0.083333333333333 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.85e-18) || !(z <= 4.8e-19)) {
		tmp = (y / (x / (z * z))) - x;
	} else {
		tmp = 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 <= (-2.85d-18)) .or. (.not. (z <= 4.8d-19))) then
        tmp = (y / (x / (z * z))) - x
    else
        tmp = 0.083333333333333d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.85e-18) || !(z <= 4.8e-19)) {
		tmp = (y / (x / (z * z))) - x;
	} else {
		tmp = 0.083333333333333 / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.85e-18) or not (z <= 4.8e-19):
		tmp = (y / (x / (z * z))) - x
	else:
		tmp = 0.083333333333333 / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.85e-18) || !(z <= 4.8e-19))
		tmp = Float64(Float64(y / Float64(x / Float64(z * z))) - x);
	else
		tmp = Float64(0.083333333333333 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.85e-18) || ~((z <= 4.8e-19)))
		tmp = (y / (x / (z * z))) - x;
	else
		tmp = 0.083333333333333 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.85e-18], N[Not[LessEqual[z, 4.8e-19]], $MachinePrecision]], N[(N[(y / N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(0.083333333333333 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.84999999999999986e-18 or 4.80000000000000046e-19 < z

   1. Initial program 89.2%

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

   2. Taylor expanded in y around inf 64.3%

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

      1. associate-/l* 68.6%

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

      2. unpow2 68.6%

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

   4. Simplified 68.6%

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

   5. Taylor expanded in x around 0 48.2%

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

   6. Taylor expanded in x around inf 48.2%

      \[\leadsto \color{blue}{-1 \cdot x} + \frac{y}{\frac{x}{z \cdot z}} \]
   7. Step-by-step derivation

      1. mul-1-neg 48.2%

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

   8. Simplified 48.2%

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

   if -2.84999999999999986e-18 < z < 4.80000000000000046e-19

   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 96.4%

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

   3. Taylor expanded in x around 0 46.0%

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

   4. Taylor expanded in x around 0 46.2%

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-18} \lor \neg \left(z \leq 4.8 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{y}{\frac{x}{z \cdot z}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \end{array} \]

Alternative 12: 23.9% 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 94.2%

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

2. Taylor expanded in z around 0 60.7%

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

3. Taylor expanded in x around 0 23.8%

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

4. Taylor expanded in x around 0 24.2%

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

5. Final simplification 24.2%

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

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 2023171 
(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)))