Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (- (fma (log y) (- -0.5 y) y) z)))

C

double code(double x, double y, double z) {
	return x + (fma(log(y), (-0.5 - y), y) - z);
}

Julia

function code(x, y, z)
	return Float64(x + Float64(fma(log(y), Float64(-0.5 - y), y) - z))
end

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. sub-neg 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+l+ 99.8%

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

   5. sub-neg 99.8%

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

   6. neg-sub0 99.8%

      \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

   7. associate-+l- 99.8%

      \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

   8. neg-sub0 99.8%

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

   9. neg-mul-1 99.8%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right) \]

Alternative 2: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-128}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+51}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e-206)
   (- x z)
   (if (<= y 1.4e-128)
     (+ x (* (log y) -0.5))
     (if (<= y 1.85e+51) (- x z) (+ x (* y (- 1.0 (log y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e-206) {
		tmp = x - z;
	} else if (y <= 1.4e-128) {
		tmp = x + (log(y) * -0.5);
	} else if (y <= 1.85e+51) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - log(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 (y <= 1.1d-206) then
        tmp = x - z
    else if (y <= 1.4d-128) then
        tmp = x + (log(y) * (-0.5d0))
    else if (y <= 1.85d+51) then
        tmp = x - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e-206) {
		tmp = x - z;
	} else if (y <= 1.4e-128) {
		tmp = x + (Math.log(y) * -0.5);
	} else if (y <= 1.85e+51) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.1e-206:
		tmp = x - z
	elif y <= 1.4e-128:
		tmp = x + (math.log(y) * -0.5)
	elif y <= 1.85e+51:
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e-206)
		tmp = Float64(x - z);
	elseif (y <= 1.4e-128)
		tmp = Float64(x + Float64(log(y) * -0.5));
	elseif (y <= 1.85e+51)
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.1e-206)
		tmp = x - z;
	elseif (y <= 1.4e-128)
		tmp = x + (log(y) * -0.5);
	elseif (y <= 1.85e+51)
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.1e-206], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.4e-128], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+51], N[(x - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.0999999999999999e-206 or 1.3999999999999999e-128 < y < 1.8500000000000001e51

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. distribute-rgt-neg-in 81.1%

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

      3. log-rec 81.1%

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

      4. remove-double-neg 81.1%

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

   4. Simplified 81.1%

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

   5. Taylor expanded in y around 0 76.0%

      \[\leadsto \color{blue}{x - z} \]

   if 1.0999999999999999e-206 < y < 1.3999999999999999e-128

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-def 100.0%

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

      8. neg-sub0 100.0%

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, x + \left(y - z\right)\right) \]

      9. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, x + \left(y - z\right)\right) \]

      10. associate--r+ 100.0%

          \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, x + \left(y - z\right)\right) \]

      11. metadata-eval 100.0%

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

      12. associate-+r- 100.0%

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

      13. +-commutative 100.0%

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

      14. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 76.0%

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

   5. Taylor expanded in y around 0 76.0%

      \[\leadsto \color{blue}{-0.5 \cdot \log y + x} \]

   if 1.8500000000000001e51 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. log-rec 99.6%

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

      4. remove-double-neg 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.7%

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

   6. Taylor expanded in z around 0 87.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-128}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+51}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.3) (- (- x (* (log y) 0.5)) z) (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - log(y))) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.3d0) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.3:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.3)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.3)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.3], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999998

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 98.6%

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

   if 3.2999999999999998 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 99.7%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 98.8%

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

      1. log-rec 98.8%

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

      2. sub-neg 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

2. Final simplification 99.8%

   \[\leadsto \left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. associate-+l- 99.8%

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

   2. *-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \left(x + \left(y - \log y \cdot \left(y + 0.5\right)\right)\right) - z \]

Alternative 6: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -510 \lor \neg \left(z \leq 112\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -510.0) (not (<= z 112.0))) (- x z) (+ x (* (log y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -510.0) || !(z <= 112.0)) {
		tmp = x - z;
	} else {
		tmp = x + (log(y) * -0.5);
	}
	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 <= (-510.0d0)) .or. (.not. (z <= 112.0d0))) then
        tmp = x - z
    else
        tmp = x + (log(y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -510.0) || !(z <= 112.0)) {
		tmp = x - z;
	} else {
		tmp = x + (Math.log(y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -510.0) or not (z <= 112.0):
		tmp = x - z
	else:
		tmp = x + (math.log(y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -510.0) || !(z <= 112.0))
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(log(y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -510.0) || ~((z <= 112.0)))
		tmp = x - z;
	else
		tmp = x + (log(y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -510.0], N[Not[LessEqual[z, 112.0]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -510 or 112 < z

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-neg-in 99.5%

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

      3. log-rec 99.5%

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

      4. remove-double-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 77.5%

      \[\leadsto \color{blue}{x - z} \]

   if -510 < z < 112

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.8%

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

      8. neg-sub0 99.8%

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, x + \left(y - z\right)\right) \]

      9. +-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, x + \left(y - z\right)\right) \]

      10. associate--r+ 99.8%

          \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, x + \left(y - z\right)\right) \]

      11. metadata-eval 99.8%

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

      12. associate-+r- 99.8%

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

      13. +-commutative 99.8%

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

      14. associate-+r- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 98.8%

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

   5. Taylor expanded in y around 0 68.4%

      \[\leadsto \color{blue}{-0.5 \cdot \log y + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -510 \lor \neg \left(z \leq 112\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \end{array} \]

Alternative 7: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.15e+51) (- (- x (* (log y) 0.5)) z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.15e+51) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + (y * (1.0 - log(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 (y <= 2.15d+51) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.15e+51) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.15e+51:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.15e+51)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.15e+51)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.15e+51], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1499999999999999e51

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 94.7%

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

   if 2.1499999999999999e51 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. log-rec 99.6%

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

      4. remove-double-neg 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.7%

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

   6. Taylor expanded in z around 0 87.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 8: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+144}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e+144) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+144) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(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 (y <= 1.35d+144) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+144) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.35e+144:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e+144)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.35e+144)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e+144], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000008e144

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. distribute-rgt-neg-in 80.3%

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

      3. log-rec 80.3%

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

      4. remove-double-neg 80.3%

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

   4. Simplified 80.3%

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

   5. Taylor expanded in y around 0 69.2%

      \[\leadsto \color{blue}{x - z} \]

   if 1.35000000000000008e144 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-neg-in 99.5%

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

      3. log-rec 99.5%

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

      4. remove-double-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.7%

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

   6. Taylor expanded in z around 0 90.0%

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

      1. *-commutative 90.0%

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

      2. fma-udef 90.0%

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

   8. Simplified 90.0%

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

   9. Taylor expanded in x around 0 73.5%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+144}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 9: 48.5% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+89}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.5e+89) (- z) (if (<= z 5e+42) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e+89) {
		tmp = -z;
	} else if (z <= 5e+42) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-7.5d+89)) then
        tmp = -z
    else if (z <= 5d+42) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e+89) {
		tmp = -z;
	} else if (z <= 5e+42) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.5e+89:
		tmp = -z
	elif z <= 5e+42:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.5e+89)
		tmp = Float64(-z);
	elseif (z <= 5e+42)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.5e+89)
		tmp = -z;
	elseif (z <= 5e+42)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.5e+89], (-z), If[LessEqual[z, 5e+42], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -7.49999999999999947e89 or 5.00000000000000007e42 < z

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate-+l- 99.8%

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

      2. *-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in z around inf 66.9%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   5. Step-by-step derivation

      1. neg-mul-1 66.9%

         \[\leadsto \color{blue}{-z} \]

   6. Simplified 66.9%

      \[\leadsto \color{blue}{-z} \]

   if -7.49999999999999947e89 < z < 5.00000000000000007e42

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

      5. sub-neg 99.9%

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

      6. neg-sub0 99.9%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 99.9%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 99.9%

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

      9. neg-mul-1 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 44.7%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+89}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+42}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10: 58.0% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ x - z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

2. Taylor expanded in y around inf 84.7%

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

   1. mul-1-neg 84.7%

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

   2. distribute-rgt-neg-in 84.7%

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

   3. log-rec 84.7%

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

   4. remove-double-neg 84.7%

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

4. Simplified 84.7%

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

5. Taylor expanded in y around 0 59.0%

   \[\leadsto \color{blue}{x - z} \]

6. Final simplification 59.0%

   \[\leadsto x - z \]

Alternative 11: 30.2% accurate, 111.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. sub-neg 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+l+ 99.8%

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

   5. sub-neg 99.8%

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

   6. neg-sub0 99.8%

      \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

   7. associate-+l- 99.8%

      \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

   8. neg-sub0 99.8%

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

   9. neg-mul-1 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 33.3%

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

5. Final simplification 33.3%

   \[\leadsto x \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023258 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))