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

Percentage Accurate: 94.4% → 99.5%

Time: 17.8s

Alternatives: 11

Speedup: 1.0×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, -x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 200000000.0)
   (+
    (fma (+ x -0.5) (log x) (- x))
    (+
     0.91893853320467
     (/
      (fma
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       z
       0.083333333333333)
      x)))
   (+
    (- (+ 0.91893853320467 (* (+ x -0.5) (log x))) x)
    (* (+ y 0.0007936500793651) (* z (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 200000000.0) {
		tmp = fma((x + -0.5), log(x), -x) + (0.91893853320467 + (fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x));
	} else {
		tmp = ((0.91893853320467 + ((x + -0.5) * log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 200000000.0)
		tmp = Float64(fma(Float64(x + -0.5), log(x), Float64(-x)) + Float64(0.91893853320467 + Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x)));
	else
		tmp = Float64(Float64(Float64(0.91893853320467 + Float64(Float64(x + -0.5) * log(x))) - x) + Float64(Float64(y + 0.0007936500793651) * Float64(z * Float64(z / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 200000000.0], N[(N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.91893853320467 + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e8

   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. associate-+l+ 99.7%

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

      2. fma-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. fma-def 99.7%

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

      6. fma-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   if 2e8 < x

   1. Initial program 85.6%

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

      1. sub-neg 85.6%

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

      2. associate-+l+ 85.6%

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

      3. +-commutative 85.6%

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

      4. sub-neg 85.6%

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

      5. associate-+r- 85.6%

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

      6. sub-neg 85.6%

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

      7. metadata-eval 85.6%

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

      8. *-commutative 85.6%

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

   4. Applied egg-rr 85.6%

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

   5. Taylor expanded in z around inf 85.6%

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

      1. associate-/l* 89.2%

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

      2. +-commutative 89.2%

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

      3. associate-/r/ 89.2%

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

      4. +-commutative 89.2%

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

   7. Simplified 89.2%

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

      1. unpow2 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. times-frac 99.6%

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

   9. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log x \cdot \left(x + -0.5\right) + 0.91893853320467\right) - x\right) + \color{blue}{\left(\frac{z}{1} \cdot \frac{z}{x}\right)} \cdot \left(0.0007936500793651 + y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(x + -0.5, \log x, -x\right) + \left(0.91893853320467 + \frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 200000000:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 200000000.0)
   (+
    (/
     (+
      0.083333333333333
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778)))
     x)
    (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x)))
   (+
    (- (+ 0.91893853320467 (* (+ x -0.5) (log x))) x)
    (* (+ y 0.0007936500793651) (* z (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 200000000.0) {
		tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	} else {
		tmp = ((0.91893853320467 + ((x + -0.5) * log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / 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 <= 200000000.0d0) then
        tmp = ((0.083333333333333d0 + (z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0))) / x) + (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x))
    else
        tmp = ((0.91893853320467d0 + ((x + (-0.5d0)) * log(x))) - x) + ((y + 0.0007936500793651d0) * (z * (z / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 200000000.0) {
		tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x));
	} else {
		tmp = ((0.91893853320467 + ((x + -0.5) * Math.log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 200000000.0:
		tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x))
	else:
		tmp = ((0.91893853320467 + ((x + -0.5) * math.log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 200000000.0)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)));
	else
		tmp = Float64(Float64(Float64(0.91893853320467 + Float64(Float64(x + -0.5) * log(x))) - x) + Float64(Float64(y + 0.0007936500793651) * Float64(z * Float64(z / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 200000000.0)
		tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (0.91893853320467 + ((log(x) * (x - 0.5)) - x));
	else
		tmp = ((0.91893853320467 + ((x + -0.5) * log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 200000000.0], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.91893853320467 + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e8

   1. Initial program 99.7%

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

   if 2e8 < x

   1. Initial program 85.6%

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

      1. sub-neg 85.6%

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

      2. associate-+l+ 85.6%

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

      3. +-commutative 85.6%

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

      4. sub-neg 85.6%

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

      5. associate-+r- 85.6%

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

      6. sub-neg 85.6%

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

      7. metadata-eval 85.6%

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

      8. *-commutative 85.6%

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

   4. Applied egg-rr 85.6%

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

   5. Taylor expanded in z around inf 85.6%

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

      1. associate-/l* 89.2%

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

      2. +-commutative 89.2%

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

      3. associate-/r/ 89.2%

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

      4. +-commutative 89.2%

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

   7. Simplified 89.2%

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

      1. unpow2 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. times-frac 99.6%

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

   9. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log x \cdot \left(x + -0.5\right) + 0.91893853320467\right) - x\right) + \color{blue}{\left(\frac{z}{1} \cdot \frac{z}{x}\right)} \cdot \left(0.0007936500793651 + y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200000000:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\\ \mathbf{if}\;x \leq 50000:\\ \;\;\;\;t\_0 + \frac{0.083333333333333 + z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ 0.91893853320467 (* (+ x -0.5) (log x))) x)))
   (if (<= x 50000.0)
     (+
      t_0
      (/
       (+
        0.083333333333333
        (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778)))
       x))
     (+ t_0 (* (+ y 0.0007936500793651) (* z (/ z x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((x + -0.5) * log(x))) - x;
	double tmp;
	if (x <= 50000.0) {
		tmp = t_0 + ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x);
	} else {
		tmp = t_0 + ((y + 0.0007936500793651) * (z * (z / 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 + ((x + (-0.5d0)) * log(x))) - x
    if (x <= 50000.0d0) then
        tmp = t_0 + ((0.083333333333333d0 + (z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0))) / x)
    else
        tmp = t_0 + ((y + 0.0007936500793651d0) * (z * (z / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (0.91893853320467 + ((x + -0.5) * Math.log(x))) - x;
	double tmp;
	if (x <= 50000.0) {
		tmp = t_0 + ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x);
	} else {
		tmp = t_0 + ((y + 0.0007936500793651) * (z * (z / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.91893853320467 + ((x + -0.5) * math.log(x))) - x
	tmp = 0
	if x <= 50000.0:
		tmp = t_0 + ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x)
	else:
		tmp = t_0 + ((y + 0.0007936500793651) * (z * (z / x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.91893853320467 + Float64(Float64(x + -0.5) * log(x))) - x)
	tmp = 0.0
	if (x <= 50000.0)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778))) / x));
	else
		tmp = Float64(t_0 + Float64(Float64(y + 0.0007936500793651) * Float64(z * Float64(z / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (0.91893853320467 + ((x + -0.5) * log(x))) - x;
	tmp = 0.0;
	if (x <= 50000.0)
		tmp = t_0 + ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x);
	else
		tmp = t_0 + ((y + 0.0007936500793651) * (z * (z / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(0.91893853320467 + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, 50000.0], N[(t$95$0 + N[(N[(0.083333333333333 + N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e4

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+r- 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 5e4 < x

   1. Initial program 85.6%

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

      1. sub-neg 85.6%

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

      2. associate-+l+ 85.6%

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

      3. +-commutative 85.6%

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

      4. sub-neg 85.6%

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

      5. associate-+r- 85.6%

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

      6. sub-neg 85.6%

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

      7. metadata-eval 85.6%

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

      8. *-commutative 85.6%

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

   4. Applied egg-rr 85.6%

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

   5. Taylor expanded in z around inf 85.6%

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

      1. associate-/l* 89.2%

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

      2. +-commutative 89.2%

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

      3. associate-/r/ 89.2%

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

      4. +-commutative 89.2%

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

   7. Simplified 89.2%

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

      1. unpow2 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. times-frac 99.6%

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

   9. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log x \cdot \left(x + -0.5\right) + 0.91893853320467\right) - x\right) + \color{blue}{\left(\frac{z}{1} \cdot \frac{z}{x}\right)} \cdot \left(0.0007936500793651 + y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50000:\\ \;\;\;\;\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \frac{0.083333333333333 + z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.192:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + -0.5 \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.192)
   (+
    (/
     (+
      0.083333333333333
      (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778)))
     x)
    (+ 0.91893853320467 (* -0.5 (log x))))
   (+
    (- (+ 0.91893853320467 (* (+ x -0.5) (log x))) x)
    (* (+ y 0.0007936500793651) (* z (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.192) {
		tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (0.91893853320467 + (-0.5 * log(x)));
	} else {
		tmp = ((0.91893853320467 + ((x + -0.5) * log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / 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 <= 0.192d0) then
        tmp = ((0.083333333333333d0 + (z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0))) / x) + (0.91893853320467d0 + ((-0.5d0) * log(x)))
    else
        tmp = ((0.91893853320467d0 + ((x + (-0.5d0)) * log(x))) - x) + ((y + 0.0007936500793651d0) * (z * (z / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.192) {
		tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (0.91893853320467 + (-0.5 * Math.log(x)));
	} else {
		tmp = ((0.91893853320467 + ((x + -0.5) * Math.log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.192:
		tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (0.91893853320467 + (-0.5 * math.log(x)))
	else:
		tmp = ((0.91893853320467 + ((x + -0.5) * math.log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.192)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + Float64(0.91893853320467 + Float64(-0.5 * log(x))));
	else
		tmp = Float64(Float64(Float64(0.91893853320467 + Float64(Float64(x + -0.5) * log(x))) - x) + Float64(Float64(y + 0.0007936500793651) * Float64(z * Float64(z / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.192)
		tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (0.91893853320467 + (-0.5 * log(x)));
	else
		tmp = ((0.91893853320467 + ((x + -0.5) * log(x))) - x) + ((y + 0.0007936500793651) * (z * (z / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.192], N[(N[(N[(0.083333333333333 + N[(z * N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 + N[(-0.5 * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.91893853320467 + N[(N[(x + -0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.192

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   if 0.192 < x

   1. Initial program 86.0%

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

      1. sub-neg 86.0%

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

      2. associate-+l+ 86.0%

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

      3. +-commutative 86.0%

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

      4. sub-neg 86.0%

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

      5. associate-+r- 86.0%

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

      6. sub-neg 86.0%

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

      7. metadata-eval 86.0%

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

      8. *-commutative 86.0%

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

   4. Applied egg-rr 86.0%

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

   5. Taylor expanded in z around inf 85.9%

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

      1. associate-/l* 89.4%

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

      2. +-commutative 89.4%

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

      3. associate-/r/ 89.4%

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

      4. +-commutative 89.4%

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

   7. Simplified 89.4%

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

      1. unpow2 89.4%

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

      2. *-un-lft-identity 89.4%

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

      3. times-frac 99.4%

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

   9. Applied egg-rr 99.4%

      \[\leadsto \left(\left(\log x \cdot \left(x + -0.5\right) + 0.91893853320467\right) - x\right) + \color{blue}{\left(\frac{z}{1} \cdot \frac{z}{x}\right)} \cdot \left(0.0007936500793651 + y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.192:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)}{x} + \left(0.91893853320467 + -0.5 \cdot \log x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \left(y + 0.0007936500793651\right) \cdot \left(z \cdot \frac{z}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333 + z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)}{x} + x \cdot \left(\log x + -1\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (/
   (+
    0.083333333333333
    (* z (- (* (+ y 0.0007936500793651) z) 0.0027777777777778)))
   x)
  (* x (+ (log x) -1.0))))

C

double code(double x, double y, double z) {
	return ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (x * (log(x) + -1.0));
}

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 + (z * (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0))) / x) + (x * (log(x) + (-1.0d0)))
end function

Java

public static double code(double x, double y, double z) {
	return ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (x * (Math.log(x) + -1.0));
}

Python

def code(x, y, z):
	return ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (x * (math.log(x) + -1.0))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + Float64(x * Float64(log(x) + -1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((0.083333333333333 + (z * (((y + 0.0007936500793651) * z) - 0.0027777777777778))) / x) + (x * (log(x) + -1.0));
end

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in x around inf 91.9%

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

   1. sub-neg 52.4%

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

   2. mul-1-neg 52.4%

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

   3. log-rec 52.4%

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

   4. remove-double-neg 52.4%

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

   5. metadata-eval 52.4%

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

5. Simplified 91.9%

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

6. Final simplification 91.9%

   \[\leadsto \frac{0.083333333333333 + z \cdot \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right)}{x} + x \cdot \left(\log x + -1\right) \]
7. Add Preprocessing

Alternative 6: 57.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{1}{x \cdot 12.000000000000048} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ 0.91893853320467 (- (* (log x) (- x 0.5)) x))
  (/ 1.0 (* x 12.000000000000048))))

C

double code(double x, double y, double z) {
	return (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (1.0 / (x * 12.000000000000048));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (0.91893853320467d0 + ((log(x) * (x - 0.5d0)) - x)) + (1.0d0 / (x * 12.000000000000048d0))
end function

Java

public static double code(double x, double y, double z) {
	return (0.91893853320467 + ((Math.log(x) * (x - 0.5)) - x)) + (1.0 / (x * 12.000000000000048));
}

Python

def code(x, y, z):
	return (0.91893853320467 + ((math.log(x) * (x - 0.5)) - x)) + (1.0 / (x * 12.000000000000048))

Julia

function code(x, y, z)
	return Float64(Float64(0.91893853320467 + Float64(Float64(log(x) * Float64(x - 0.5)) - x)) + Float64(1.0 / Float64(x * 12.000000000000048)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.91893853320467 + ((log(x) * (x - 0.5)) - x)) + (1.0 / (x * 12.000000000000048));
end

Wolfram

code[x_, y_, z_] := N[(N[(0.91893853320467 + N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * 12.000000000000048), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 52.8%

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

   1. clear-num 52.8%

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

   2. inv-pow 52.8%

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

5. Applied egg-rr 52.8%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{{\left(\frac{x}{0.083333333333333}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 52.8%

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

7. Simplified 52.8%

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

8. Taylor expanded in x around 0 52.9%

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{1}{\color{blue}{12.000000000000048 \cdot x}} \]
9. Step-by-step derivation

   1. *-commutative 52.9%

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

10. Simplified 52.9%

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

11. Final simplification 52.9%

    \[\leadsto \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{1}{x \cdot 12.000000000000048} \]
12. Add Preprocessing

Alternative 7: 57.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 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. Initial program 92.4%

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

3. Taylor expanded in z around 0 52.8%

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

4. Final simplification 52.8%

   \[\leadsto \left(0.91893853320467 + \left(\log x \cdot \left(x - 0.5\right) - x\right)\right) + \frac{0.083333333333333}{x} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

   1. sub-neg 92.4%

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

   2. associate-+l+ 92.4%

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

   3. +-commutative 92.4%

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

   4. sub-neg 92.4%

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

   5. associate-+r- 92.4%

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

   6. sub-neg 92.4%

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

   7. metadata-eval 92.4%

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

   8. *-commutative 92.4%

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

4. Applied egg-rr 92.4%

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

5. Taylor expanded in z around 0 52.8%

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

6. Final simplification 52.8%

   \[\leadsto \left(\left(0.91893853320467 + \left(x + -0.5\right) \cdot \log x\right) - x\right) + \frac{0.083333333333333}{x} \]
7. Add Preprocessing

Alternative 9: 56.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (log(x) + (-1.0d0))) + (0.083333333333333d0 / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 52.8%

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

4. Taylor expanded in x around inf 52.4%

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

   1. sub-neg 52.4%

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

   2. mul-1-neg 52.4%

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

   3. log-rec 52.4%

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

   4. remove-double-neg 52.4%

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

   5. metadata-eval 52.4%

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

6. Simplified 52.4%

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

7. Final simplification 52.4%

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

Alternative 10: 23.3% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ 0.083333333333333 \cdot \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 * (1.0d0 / x)
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 * (1.0 / x);
}

Python

def code(x, y, z):
	return 0.083333333333333 * (1.0 / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around 0 52.8%

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

4. Taylor expanded in x around inf 52.4%

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

   1. sub-neg 52.4%

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

   2. mul-1-neg 52.4%

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

   3. log-rec 52.4%

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

   4. remove-double-neg 52.4%

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

   5. metadata-eval 52.4%

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

6. Simplified 52.4%

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

7. Taylor expanded in x around 0 22.2%

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

   1. div-inv 22.2%

      \[\leadsto \color{blue}{0.083333333333333 \cdot \frac{1}{x}} \]

9. Applied egg-rr 22.2%

   \[\leadsto \color{blue}{0.083333333333333 \cdot \frac{1}{x}} \]

10. Final simplification 22.2%

    \[\leadsto 0.083333333333333 \cdot \frac{1}{x} \]
11. Add Preprocessing

Alternative 11: 23.3% 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 92.4%

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

3. Taylor expanded in z around 0 52.8%

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

4. Taylor expanded in x around inf 52.4%

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

   1. sub-neg 52.4%

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

   2. mul-1-neg 52.4%

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

   3. log-rec 52.4%

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

   4. remove-double-neg 52.4%

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

   5. metadata-eval 52.4%

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

6. Simplified 52.4%

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

7. Taylor expanded in x around 0 22.2%

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

8. Final simplification 22.2%

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

Developer target: 98.7% 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 2024096 
(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)))