Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.7s

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. 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. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-138}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+100}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e-138)
   (- (* (log y) -0.5) z)
   (if (<= y 1.35e+100) (- x z) (+ x (- y (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-138) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.35e+100) {
		tmp = x - z;
	} else {
		tmp = x + (y - (y * 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 <= 7d-138) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.35d+100) then
        tmp = x - z
    else
        tmp = x + (y - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-138) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.35e+100) {
		tmp = x - z;
	} else {
		tmp = x + (y - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7e-138:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.35e+100:
		tmp = x - z
	else:
		tmp = x + (y - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7e-138)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.35e+100)
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7e-138)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.35e+100)
		tmp = x - z;
	else
		tmp = x + (y - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7e-138], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.35e+100], N[(x - z), $MachinePrecision], N[(x + N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.9999999999999997e-138

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in x around 0 75.2%

      \[\leadsto \color{blue}{-1 \cdot \left(z + 0.5 \cdot \log y\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 75.2%

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

      2. distribute-neg-in 75.2%

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

      3. sub-neg 75.2%

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

      4. neg-sub0 75.2%

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

      5. associate--r+ 75.2%

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

      6. +-commutative 75.2%

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

      7. associate--r+ 75.2%

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

      8. neg-sub0 75.2%

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

      9. *-commutative 75.2%

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

      10. distribute-rgt-neg-in 75.2%

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

      11. metadata-eval 75.2%

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

   6. Simplified 75.2%

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

   if 6.9999999999999997e-138 < y < 1.34999999999999999e100

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 84.6%

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

   4. Taylor expanded in z around inf 74.9%

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

   if 1.34999999999999999e100 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 86.1%

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

      1. associate-*r* 86.1%

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

      2. neg-mul-1 86.1%

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

      3. +-commutative 86.1%

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

      4. cancel-sign-sub-inv 86.1%

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

   7. Simplified 86.1%

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

   8. Taylor expanded in y around inf 86.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-138}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+100}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-138}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+100}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.3e-138)
   (- (* (log y) -0.5) z)
   (if (<= y 1.15e+100) (- x z) (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e-138) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.15e+100) {
		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 <= 2.3d-138) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.15d+100) 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 <= 2.3e-138) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.15e+100) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.3e-138:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.15e+100:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.3e-138)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.15e+100)
		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 <= 2.3e-138)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.15e+100)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.3e-138], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.15e+100], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.2999999999999999e-138

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in x around 0 75.2%

      \[\leadsto \color{blue}{-1 \cdot \left(z + 0.5 \cdot \log y\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 75.2%

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

      2. distribute-neg-in 75.2%

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

      3. sub-neg 75.2%

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

      4. neg-sub0 75.2%

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

      5. associate--r+ 75.2%

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

      6. +-commutative 75.2%

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

      7. associate--r+ 75.2%

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

      8. neg-sub0 75.2%

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

      9. *-commutative 75.2%

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

      10. distribute-rgt-neg-in 75.2%

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

      11. metadata-eval 75.2%

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

   6. Simplified 75.2%

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

   if 2.2999999999999999e-138 < y < 1.14999999999999995e100

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 84.6%

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

   4. Taylor expanded in z around inf 74.9%

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

   if 1.14999999999999995e100 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 74.0%

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

      1. log-rec 74.0%

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

      2. sub-neg 74.0%

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

   7. Simplified 74.0%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(y + \left(x - \log y \cdot \left(\left(y + 1.5\right) + -1\right)\right)\right) - z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (+ y (- x (* (log y) (+ (+ y 1.5) -1.0)))) z))

C

double code(double x, double y, double z) {
	return (y + (x - (log(y) * ((y + 1.5) + -1.0)))) - 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 + 1.5d0) + (-1.0d0))))) - z
end function

Java

public static double code(double x, double y, double z) {
	return (y + (x - (Math.log(y) * ((y + 1.5) + -1.0)))) - z;
}

Python

def code(x, y, z):
	return (y + (x - (math.log(y) * ((y + 1.5) + -1.0)))) - z

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(x - Float64(log(y) * Float64(Float64(y + 1.5) + -1.0)))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + (x - (log(y) * ((y + 1.5) + -1.0)))) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(N[(y + 1.5), $MachinePrecision] + -1.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 96.9%

      \[\leadsto \left(\left(x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y + 0.5\right)\right)} \cdot \log y\right) + y\right) - z \]

   2. expm1-undefine 96.9%

      \[\leadsto \left(\left(x - \color{blue}{\left(e^{\mathsf{log1p}\left(y + 0.5\right)} - 1\right)} \cdot \log y\right) + y\right) - z \]

4. Applied egg-rr 96.9%

   \[\leadsto \left(\left(x - \color{blue}{\left(e^{\mathsf{log1p}\left(y + 0.5\right)} - 1\right)} \cdot \log y\right) + y\right) - z \]
5. Step-by-step derivation

   1. sub-neg 96.9%

      \[\leadsto \left(\left(x - \color{blue}{\left(e^{\mathsf{log1p}\left(y + 0.5\right)} + \left(-1\right)\right)} \cdot \log y\right) + y\right) - z \]

   2. log1p-undefine 96.9%

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

   3. rem-exp-log 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. metadata-eval 99.8%

      \[\leadsto \left(\left(x - \left(\left(\color{blue}{1.5} + y\right) + \left(-1\right)\right) \cdot \log y\right) + y\right) - z \]

   7. metadata-eval 99.8%

      \[\leadsto \left(\left(x - \left(\left(1.5 + y\right) + \color{blue}{-1}\right) \cdot \log y\right) + y\right) - z \]

6. Simplified 99.8%

   \[\leadsto \left(\left(x - \color{blue}{\left(\left(1.5 + y\right) + -1\right)} \cdot \log y\right) + y\right) - z \]

7. Final simplification 99.8%

   \[\leadsto \left(y + \left(x - \log y \cdot \left(\left(y + 1.5\right) + -1\right)\right)\right) - z \]
8. Add Preprocessing

Alternative 5: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 880000:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 880000.0) (- x (+ z (* (log y) 0.5))) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 880000.0) {
		tmp = x - (z + (log(y) * 0.5));
	} else {
		tmp = (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 <= 880000.0d0) then
        tmp = x - (z + (log(y) * 0.5d0))
    else
        tmp = (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 <= 880000.0) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 880000.0:
		tmp = x - (z + (math.log(y) * 0.5))
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 880000.0)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	else
		tmp = 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 <= 880000.0)
		tmp = x - (z + (log(y) * 0.5));
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 880000.0], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.8e5

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.4%

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

   if 8.8e5 < y

   1. Initial program 99.6%

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

      1. expm1-log1p-u 94.0%

         \[\leadsto \left(\left(x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y + 0.5\right)\right)} \cdot \log y\right) + y\right) - z \]

      2. expm1-undefine 94.0%

         \[\leadsto \left(\left(x - \color{blue}{\left(e^{\mathsf{log1p}\left(y + 0.5\right)} - 1\right)} \cdot \log y\right) + y\right) - z \]

   4. Applied egg-rr 94.0%

      \[\leadsto \left(\left(x - \color{blue}{\left(e^{\mathsf{log1p}\left(y + 0.5\right)} - 1\right)} \cdot \log y\right) + y\right) - z \]
   5. Step-by-step derivation

      1. sub-neg 94.0%

         \[\leadsto \left(\left(x - \color{blue}{\left(e^{\mathsf{log1p}\left(y + 0.5\right)} + \left(-1\right)\right)} \cdot \log y\right) + y\right) - z \]

      2. log1p-undefine 94.0%

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

      3. rem-exp-log 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. metadata-eval 99.6%

         \[\leadsto \left(\left(x - \left(\left(\color{blue}{1.5} + y\right) + \left(-1\right)\right) \cdot \log y\right) + y\right) - z \]

      7. metadata-eval 99.6%

         \[\leadsto \left(\left(x - \left(\left(1.5 + y\right) + \color{blue}{-1}\right) \cdot \log y\right) + y\right) - z \]

   6. Simplified 99.6%

      \[\leadsto \left(\left(x - \color{blue}{\left(\left(1.5 + y\right) + -1\right)} \cdot \log y\right) + y\right) - z \]

   7. Taylor expanded in y around inf 82.6%

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

      1. mul-1-neg 82.6%

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

      2. log-rec 82.6%

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

      3. remove-double-neg 82.6%

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

   9. Simplified 82.6%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 880000:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+99}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.4e+99) (- x (+ z (* (log y) 0.5))) (+ x (- y (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+99) {
		tmp = x - (z + (log(y) * 0.5));
	} else {
		tmp = x + (y - (y * 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 <= 6.4d+99) then
        tmp = x - (z + (log(y) * 0.5d0))
    else
        tmp = x + (y - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+99) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else {
		tmp = x + (y - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.4e+99:
		tmp = x - (z + (math.log(y) * 0.5))
	else:
		tmp = x + (y - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.4e+99)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	else
		tmp = Float64(x + Float64(y - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.4e+99)
		tmp = x - (z + (log(y) * 0.5));
	else
		tmp = x + (y - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.4e+99], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.39999999999999999e99

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 91.5%

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

   if 6.39999999999999999e99 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 86.1%

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

      1. associate-*r* 86.1%

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

      2. neg-mul-1 86.1%

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

      3. +-commutative 86.1%

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

      4. cancel-sign-sub-inv 86.1%

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

   7. Simplified 86.1%

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

   8. Taylor expanded in y around inf 86.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+99}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 8: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+101}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e+101) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e+101) {
		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 <= 2.8d+101) 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 <= 2.8e+101) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.8e+101:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e+101)
		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 <= 2.8e+101)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.8e+101], 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 < 2.79999999999999981e101

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 91.5%

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

   4. Taylor expanded in z around inf 72.1%

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

   if 2.79999999999999981e101 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 74.0%

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

      1. log-rec 74.0%

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

      2. sub-neg 74.0%

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

   7. Simplified 74.0%

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

Alternative 9: 47.7% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+110}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.1e+61) x (if (<= x 3e+110) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.1e+61) {
		tmp = x;
	} else if (x <= 3e+110) {
		tmp = -z;
	} else {
		tmp = 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 <= (-5.1d+61)) then
        tmp = x
    else if (x <= 3d+110) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.1e+61) {
		tmp = x;
	} else if (x <= 3e+110) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.1e+61:
		tmp = x
	elif x <= 3e+110:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.1e+61)
		tmp = x;
	elseif (x <= 3e+110)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.1e+61)
		tmp = x;
	elseif (x <= 3e+110)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.1e+61], x, If[LessEqual[x, 3e+110], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.1000000000000001e61 or 3.00000000000000007e110 < x

   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. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 68.0%

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

   if -5.1000000000000001e61 < x < 3.00000000000000007e110

   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. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 40.7%

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

      1. neg-mul-1 40.7%

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

   7. Simplified 40.7%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.8% 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. Add Preprocessing

3. Taylor expanded in y around 0 68.8%

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

4. Taylor expanded in z around inf 56.2%

   \[\leadsto x - \color{blue}{z} \]
5. Add Preprocessing

Alternative 11: 30.7% 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. 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. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 25.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 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 2024181 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- (+ y x) z) (* (+ y 1/2) (log y))))

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