Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.2s

Alternatives: 9

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 99.9%

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

   1. associate--l+ 99.9%

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

   2. sub-neg 99.9%

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

   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 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28)
   (- (- 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 <= 0.28) {
		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 <= 0.28d0) 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 <= 0.28) {
		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 <= 0.28:
		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 <= 0.28)
		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 <= 0.28)
		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, 0.28], 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 < 0.28000000000000003

   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 99.4%

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

   if 0.28000000000000003 < 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. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      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 99.7%

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

      1. log-rec 99.7%

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

      2. sub-neg 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+104}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e+104) (- (- 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 <= 3.5e+104) {
		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 <= 3.5d+104) 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 <= 3.5e+104) {
		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 <= 3.5e+104:
		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 <= 3.5e+104)
		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 <= 3.5e+104)
		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, 3.5e+104], 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 < 3.5000000000000002e104

   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 94.0%

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

   if 3.5000000000000002e104 < 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 98.7%

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

      2. expm1-undefine 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. expm1-define 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around inf 67.1%

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

      1. associate--l+ 67.1%

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

      2. +-commutative 67.1%

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

      3. associate-/l* 67.2%

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

      4. +-commutative 67.2%

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

   9. Simplified 67.2%

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

   10. Taylor expanded in y around inf 63.6%

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

       1. mul-1-neg 63.6%

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

       2. distribute-frac-neg 63.6%

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

       3. log-rec 63.6%

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

       4. remove-double-neg 63.6%

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

   12. Simplified 63.6%

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

   13. Taylor expanded in x around 0 87.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+104}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+107}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e+107) (- x z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e+107) {
		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 <= 2.1d+107) 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 <= 2.1e+107) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.1e+107:
		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 <= 2.1e+107)
		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 <= 2.1e+107)
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e+107], N[(x - 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 < 2.1e107

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. log-prod 99.9%

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

      3. pow2 99.9%

         \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)} + \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z \]

   4. Applied egg-rr 99.9%

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

      1. distribute-lft-in 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\left(\left(y + 0.5\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z \]

      2. log-pow 99.9%

         \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z \]

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

         \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)}\right)\right) + y\right) - z \]

      2. *-commutative 99.9%

         \[\leadsto \left(\left(x - \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z \]

      3. associate-*l* 99.9%

         \[\leadsto \left(\left(x - \left(\color{blue}{2 \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z \]

      4. distribute-lft1-in 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\left(2 + 1\right) \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)}\right) + y\right) - z \]

      5. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 73.7%

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

   if 2.1e107 < 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 98.7%

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

      2. expm1-undefine 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. expm1-define 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around inf 67.1%

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

      1. associate--l+ 67.1%

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

      2. +-commutative 67.1%

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

      3. associate-/l* 67.2%

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

      4. +-commutative 67.2%

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

   9. Simplified 67.2%

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

   10. Taylor expanded in y around inf 63.6%

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

       1. mul-1-neg 63.6%

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

       2. distribute-frac-neg 63.6%

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

       3. log-rec 63.6%

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

       4. remove-double-neg 63.6%

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

   12. Simplified 63.6%

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

   13. Taylor expanded in x around 0 87.0%

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

Alternative 5: 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+106}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.6e+106) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.6e+106) {
		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 <= 5.6d+106) 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 <= 5.6e+106) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.6e+106:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.6e+106)
		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 <= 5.6e+106)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.6e+106], 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 < 5.59999999999999986e106

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. log-prod 99.9%

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

      3. pow2 99.9%

         \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)} + \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z \]

   4. Applied egg-rr 99.9%

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

      1. distribute-lft-in 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\left(\left(y + 0.5\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z \]

      2. log-pow 99.9%

         \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z \]

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

         \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)}\right)\right) + y\right) - z \]

      2. *-commutative 99.9%

         \[\leadsto \left(\left(x - \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z \]

      3. associate-*l* 99.9%

         \[\leadsto \left(\left(x - \left(\color{blue}{2 \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z \]

      4. distribute-lft1-in 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\left(2 + 1\right) \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)}\right) + y\right) - z \]

      5. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 73.7%

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

   if 5.59999999999999986e106 < 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- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 87.0%

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

      1. associate-+r+ 87.0%

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

      2. +-commutative 87.0%

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

      3. associate-*r* 87.0%

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

      4. mul-1-neg 87.0%

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

      5. +-commutative 87.0%

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

      6. cancel-sign-sub-inv 87.0%

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

      7. associate--l+ 87.0%

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

   7. Simplified 87.0%

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

   8. Taylor expanded in y around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. log-rec 72.7%

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

      3. remove-double-neg 72.7%

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

   10. Simplified 72.7%

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

Alternative 7: 48.5% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+62}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+29}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.7e+62) x (if (<= x 8e+29) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.7e+62) {
		tmp = x;
	} else if (x <= 8e+29) {
		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 <= (-3.7d+62)) then
        tmp = x
    else if (x <= 8d+29) 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 <= -3.7e+62) {
		tmp = x;
	} else if (x <= 8e+29) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.7e+62:
		tmp = x
	elif x <= 8e+29:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.7e+62)
		tmp = x;
	elseif (x <= 8e+29)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.7e+62)
		tmp = x;
	elseif (x <= 8e+29)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.7e+62], x, If[LessEqual[x, 8e+29], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.70000000000000014e62 or 7.99999999999999931e29 < x

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      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 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 64.7%

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

   if -3.70000000000000014e62 < x < 7.99999999999999931e29

   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 z around inf 43.9%

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

      1. neg-mul-1 43.9%

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

   7. Simplified 43.9%

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

Alternative 8: 58.6% 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 99.9%

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

   1. add-cube-cbrt 99.9%

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

   2. log-prod 99.8%

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

   3. pow2 99.8%

      \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)} + \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z \]

4. Applied egg-rr 99.8%

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

   1. distribute-lft-in 99.8%

      \[\leadsto \left(\left(x - \color{blue}{\left(\left(y + 0.5\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{2}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z \]

   2. log-pow 99.8%

      \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z \]

6. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

      \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)}\right)\right) + y\right) - z \]

   2. *-commutative 99.8%

      \[\leadsto \left(\left(x - \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z \]

   3. associate-*l* 99.8%

      \[\leadsto \left(\left(x - \left(\color{blue}{2 \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z \]

   4. distribute-lft1-in 99.8%

      \[\leadsto \left(\left(x - \color{blue}{\left(2 + 1\right) \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)}\right) + y\right) - z \]

   5. metadata-eval 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in x around inf 58.1%

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

Alternative 9: 29.8% 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 99.9%

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

   1. associate--l+ 99.9%

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

   2. sub-neg 99.9%

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

   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 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 28.4%

   \[\leadsto \color{blue}{x} \]
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 2024137 
(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))