Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.6s

Alternatives: 11

Speedup: 0.5×

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 99.8%

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

   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. Add Preprocessing

Alternative 2: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+29} \lor \neg \left(z \leq 3.5 \cdot 10^{+37}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \log y \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.8e+29) (not (<= z 3.5e+37)))
   (- x z)
   (- (+ x y) (* (log y) (+ y 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e+29) || !(z <= 3.5e+37)) {
		tmp = x - z;
	} else {
		tmp = (x + y) - (log(y) * (y + 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.8d+29)) .or. (.not. (z <= 3.5d+37))) then
        tmp = x - z
    else
        tmp = (x + y) - (log(y) * (y + 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e+29) || !(z <= 3.5e+37)) {
		tmp = x - z;
	} else {
		tmp = (x + y) - (Math.log(y) * (y + 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.8e+29) or not (z <= 3.5e+37):
		tmp = x - z
	else:
		tmp = (x + y) - (math.log(y) * (y + 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.8e+29) || !(z <= 3.5e+37))
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(x + y) - Float64(log(y) * Float64(y + 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.8e+29) || ~((z <= 3.5e+37)))
		tmp = x - z;
	else
		tmp = (x + y) - (log(y) * (y + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.8e+29], N[Not[LessEqual[z, 3.5e+37]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e29 or 3.5e37 < z

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.9%

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

      2. pow3 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 99.9%

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

   8. Taylor expanded in y around 0 89.8%

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

   if -2.8e29 < z < 3.5e37

   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%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 97.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+29} \lor \neg \left(z \leq 3.5 \cdot 10^{+37}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \log y \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+111}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+27}:\\ \;\;\;\;y - \left(z + \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55e+111)
   (- (+ x y) (* y (log y)))
   (if (<= x 5.3e+27) (- y (+ z (* (log y) (+ y 0.5)))) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e+111) {
		tmp = (x + y) - (y * log(y));
	} else if (x <= 5.3e+27) {
		tmp = y - (z + (log(y) * (y + 0.5)));
	} else {
		tmp = x - 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 (x <= (-1.55d+111)) then
        tmp = (x + y) - (y * log(y))
    else if (x <= 5.3d+27) then
        tmp = y - (z + (log(y) * (y + 0.5d0)))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e+111) {
		tmp = (x + y) - (y * Math.log(y));
	} else if (x <= 5.3e+27) {
		tmp = y - (z + (Math.log(y) * (y + 0.5)));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.55e+111:
		tmp = (x + y) - (y * math.log(y))
	elif x <= 5.3e+27:
		tmp = y - (z + (math.log(y) * (y + 0.5)))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55e+111)
		tmp = Float64(Float64(x + y) - Float64(y * log(y)));
	elseif (x <= 5.3e+27)
		tmp = Float64(y - Float64(z + Float64(log(y) * Float64(y + 0.5))));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.55e+111)
		tmp = (x + y) - (y * log(y));
	elseif (x <= 5.3e+27)
		tmp = y - (z + (log(y) * (y + 0.5)));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55e+111], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.3e+27], N[(y - N[(z + N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55e111

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 99.8%

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

   8. Taylor expanded in z around 0 92.7%

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

      1. +-commutative 92.7%

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

   10. Simplified 92.7%

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

   if -1.55e111 < x < 5.3000000000000004e27

   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%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 95.7%

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

   if 5.3000000000000004e27 < 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%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 99.8%

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

   8. Taylor expanded in y around 0 87.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+111}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+27}:\\ \;\;\;\;y - \left(z + \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{+110}:\\ \;\;\;\;\left(y - z\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= y 1.9e+32)
     (- (+ x (* (log y) -0.5)) z)
     (if (<= y 5.9e+110) (- (- y z) t_0) (- (+ x y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (y <= 1.9e+32) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if (y <= 5.9e+110) {
		tmp = (y - z) - t_0;
	} else {
		tmp = (x + y) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * log(y)
    if (y <= 1.9d+32) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if (y <= 5.9d+110) then
        tmp = (y - z) - t_0
    else
        tmp = (x + y) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log(y);
	double tmp;
	if (y <= 1.9e+32) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if (y <= 5.9e+110) {
		tmp = (y - z) - t_0;
	} else {
		tmp = (x + y) - t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if y <= 1.9e+32:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif y <= 5.9e+110:
		tmp = (y - z) - t_0
	else:
		tmp = (x + y) - t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (y <= 1.9e+32)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif (y <= 5.9e+110)
		tmp = Float64(Float64(y - z) - t_0);
	else
		tmp = Float64(Float64(x + y) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (y <= 1.9e+32)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif (y <= 5.9e+110)
		tmp = (y - z) - t_0;
	else
		tmp = (x + y) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.9e+32], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 5.9e+110], N[(N[(y - z), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9000000000000002e32

   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 y around 0 94.5%

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

   if 1.9000000000000002e32 < y < 5.8999999999999997e110

   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%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 88.5%

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

      1. log-rec 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in z around 0 88.4%

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

      1. neg-mul-1 88.4%

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

      2. associate-+r+ 88.5%

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

      3. sub-neg 88.5%

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

      4. mul-1-neg 88.5%

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

      5. unsub-neg 88.5%

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

   10. Simplified 88.5%

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

   if 5.8999999999999997e110 < 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%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in y around inf 98.7%

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

   8. Taylor expanded in z around 0 89.8%

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

      1. +-commutative 89.8%

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

   10. Simplified 89.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{+110}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 13000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 13000000.0) (- (+ x (* (log y) -0.5)) z) (- (+ x y) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 13000000.0) {
		tmp = (x + (log(y) * -0.5)) - 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 <= 13000000.0d0) then
        tmp = (x + (log(y) * (-0.5d0))) - 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 <= 13000000.0) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 13000000.0:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 13000000.0)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(Float64(x + y) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 13000000.0)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 13000000.0], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e7

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      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 y around 0 97.3%

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

   if 1.3e7 < 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%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 98.9%

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

      2. pow3 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in y around inf 97.6%

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

   8. Taylor expanded in z around 0 80.2%

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

      1. +-commutative 80.2%

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

   10. Simplified 80.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+123}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e+123) (- (+ x (* (log y) -0.5)) z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+123) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.3d+123) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+123) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.3e+123:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e+123)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.3e+123)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.3e+123], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999993e123

   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 y around 0 88.6%

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

   if 1.29999999999999993e123 < 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 99.6%

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

      8. +-commutative 99.6%

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

      9. distribute-neg-in 99.6%

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

      10. unsub-neg 99.6%

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 74.7%

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

      1. log-rec 74.7%

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

      2. sub-neg 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+123}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x - (log(y) * (y + 0.5))) + (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 - (log(y) * (y + 0.5d0))) + (y - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+122}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.5e+122) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+122) {
		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 <= 7.5d+122) 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 <= 7.5e+122) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.5e+122:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.5e+122)
		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 <= 7.5e+122)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.5e+122], 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 < 7.5000000000000002e122

   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%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 99.5%

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

      2. pow3 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 79.6%

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

   8. Taylor expanded in y around 0 70.4%

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

   if 7.5000000000000002e122 < 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 99.6%

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

      8. +-commutative 99.6%

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

      9. distribute-neg-in 99.6%

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

      10. unsub-neg 99.6%

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 74.7%

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

      1. log-rec 74.7%

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

      2. sub-neg 74.7%

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

   7. Simplified 74.7%

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

Alternative 9: 48.5% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+29} \lor \neg \left(z \leq 5 \cdot 10^{+31}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.1e+29) (not (<= z 5e+31))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.1e+29) || !(z <= 5e+31)) {
		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 ((z <= (-2.1d+29)) .or. (.not. (z <= 5d+31))) 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 ((z <= -2.1e+29) || !(z <= 5e+31)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.1e+29) or not (z <= 5e+31):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.1e+29) || !(z <= 5e+31))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.1e+29) || ~((z <= 5e+31)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.1e+29], N[Not[LessEqual[z, 5e+31]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000002e29 or 5.00000000000000027e31 < z

   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+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      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 z around inf 68.9%

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

      1. neg-mul-1 68.9%

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

   7. Simplified 68.9%

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

   if -2.1000000000000002e29 < z < 5.00000000000000027e31

   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 99.7%

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

      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 x around inf 36.4%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+29} \lor \neg \left(z \leq 5 \cdot 10^{+31}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.3%

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

   2. pow3 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in y around inf 85.0%

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

8. Taylor expanded in y around 0 58.0%

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

Alternative 11: 30.2% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. 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 x around inf 30.3%

   \[\leadsto \color{blue}{x} \]
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 2024156 
(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))