System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2

Percentage Accurate: 99.9% → 99.9%

Time: 11.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(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 * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(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 * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 + \left(\log z - z\right), y, x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (+ 1.0 (- (log z) z)) y (* x 0.5)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. *-commutative 99.9%

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

   3. fma-def 99.9%

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

   4. sub-neg 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. sub-neg 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(1 + \left(\log z - z\right), y, x \cdot 0.5\right) \]

Alternative 2: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -20000 \lor \neg \left(x \cdot 0.5 \leq 10^{-31}\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -20000.0) (not (<= (* x 0.5) 1e-31)))
   (- (* x 0.5) (* z y))
   (+ y (* y (- (log z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -20000.0) || !((x * 0.5) <= 1e-31)) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y + (y * (log(z) - 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 * 0.5d0) <= (-20000.0d0)) .or. (.not. ((x * 0.5d0) <= 1d-31))) then
        tmp = (x * 0.5d0) - (z * y)
    else
        tmp = y + (y * (log(z) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -20000.0) || !((x * 0.5) <= 1e-31)) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y + (y * (Math.log(z) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -20000.0) or not ((x * 0.5) <= 1e-31):
		tmp = (x * 0.5) - (z * y)
	else:
		tmp = y + (y * (math.log(z) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -20000.0) || !(Float64(x * 0.5) <= 1e-31))
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	else
		tmp = Float64(y + Float64(y * Float64(log(z) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -20000.0) || ~(((x * 0.5) <= 1e-31)))
		tmp = (x * 0.5) - (z * y);
	else
		tmp = y + (y * (log(z) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -20000.0], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 1e-31]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -2e4 or 1e-31 < (*.f64 x 1/2)

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.7%

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

      2. pow3 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-+l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. sub-neg 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in z around inf 88.0%

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

      1. associate-*r* 88.0%

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

      2. neg-mul-1 88.0%

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

      3. *-commutative 88.0%

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

   6. Simplified 88.0%

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

      1. *-commutative 88.0%

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

      2. distribute-rgt-neg-out 88.0%

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

      3. unsub-neg 88.0%

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

      4. *-commutative 88.0%

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

   8. Applied egg-rr 88.0%

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

   if -2e4 < (*.f64 x 1/2) < 1e-31

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-+l+ 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. associate-+r+ 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 84.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -20000 \lor \neg \left(x \cdot 0.5 \leq 10^{-31}\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \end{array} \]

Alternative 3: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -20000 \lor \neg \left(x \cdot 0.5 \leq 10^{-31}\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -20000.0) (not (<= (* x 0.5) 1e-31)))
   (- (* x 0.5) (* z y))
   (* y (+ 1.0 (- (log z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -20000.0) || !((x * 0.5) <= 1e-31)) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y * (1.0 + (log(z) - 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 * 0.5d0) <= (-20000.0d0)) .or. (.not. ((x * 0.5d0) <= 1d-31))) then
        tmp = (x * 0.5d0) - (z * y)
    else
        tmp = y * (1.0d0 + (log(z) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -20000.0) || !((x * 0.5) <= 1e-31)) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y * (1.0 + (Math.log(z) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -20000.0) or not ((x * 0.5) <= 1e-31):
		tmp = (x * 0.5) - (z * y)
	else:
		tmp = y * (1.0 + (math.log(z) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -20000.0) || !(Float64(x * 0.5) <= 1e-31))
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	else
		tmp = Float64(y * Float64(1.0 + Float64(log(z) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -20000.0) || ~(((x * 0.5) <= 1e-31)))
		tmp = (x * 0.5) - (z * y);
	else
		tmp = y * (1.0 + (log(z) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -20000.0], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 1e-31]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -2e4 or 1e-31 < (*.f64 x 1/2)

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.7%

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

      2. pow3 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-+l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. sub-neg 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in z around inf 88.0%

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

      1. associate-*r* 88.0%

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

      2. neg-mul-1 88.0%

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

      3. *-commutative 88.0%

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

   6. Simplified 88.0%

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

      1. *-commutative 88.0%

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

      2. distribute-rgt-neg-out 88.0%

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

      3. unsub-neg 88.0%

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

      4. *-commutative 88.0%

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

   8. Applied egg-rr 88.0%

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

   if -2e4 < (*.f64 x 1/2) < 1e-31

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-+l+ 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. associate-+r+ 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around -inf 84.2%

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

      1. mul-1-neg 84.2%

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

      2. distribute-rgt-neg-in 84.2%

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

      3. sub-neg 84.2%

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

      4. mul-1-neg 84.2%

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

      5. sub-neg 84.2%

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

      6. +-commutative 84.2%

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

      7. distribute-neg-in 84.2%

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

      8. remove-double-neg 84.2%

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

      9. sub-neg 84.2%

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

      10. metadata-eval 84.2%

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

      11. +-commutative 84.2%

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

   6. Simplified 84.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -20000 \lor \neg \left(x \cdot 0.5 \leq 10^{-31}\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.28) (+ (* (log z) y) (+ y (* x 0.5))) (- (* x 0.5) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (log(z) * y) + (y + (x * 0.5));
	} else {
		tmp = (x * 0.5) - (z * 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 (z <= 0.28d0) then
        tmp = (log(z) * y) + (y + (x * 0.5d0))
    else
        tmp = (x * 0.5d0) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (Math.log(z) * y) + (y + (x * 0.5));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.28:
		tmp = (math.log(z) * y) + (y + (x * 0.5))
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.28)
		tmp = Float64(Float64(log(z) * y) + Float64(y + Float64(x * 0.5)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.28)
		tmp = (log(z) * y) + (y + (x * 0.5));
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.28], N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] + N[(y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-+l+ 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. associate-+r+ 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.0%

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

   if 0.28000000000000003 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt 99.2%

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

      2. pow3 99.1%

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

      3. sub-neg 99.1%

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

      4. associate-+l+ 99.1%

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

      5. +-commutative 99.1%

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

      6. sub-neg 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in z around inf 98.1%

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

      1. associate-*r* 98.1%

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

      2. neg-mul-1 98.1%

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

      3. *-commutative 98.1%

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

   6. Simplified 98.1%

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

      1. *-commutative 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

      3. unsub-neg 98.1%

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

      4. *-commutative 98.1%

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

   8. Applied egg-rr 98.1%

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

4. Final simplification 98.5%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (log(z) + (1.0d0 - z))) + (x * 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

2. Final simplification 99.9%

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

Alternative 6: 59.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 550000 \lor \neg \left(z \leq 6.5 \cdot 10^{+36}\right) \land \left(z \leq 2.5 \cdot 10^{+80} \lor \neg \left(z \leq 5.3 \cdot 10^{+132}\right) \land z \leq 3 \cdot 10^{+140}\right):\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 550000.0)
         (and (not (<= z 6.5e+36))
              (or (<= z 2.5e+80) (and (not (<= z 5.3e+132)) (<= z 3e+140)))))
   (* x 0.5)
   (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 550000.0) || (!(z <= 6.5e+36) && ((z <= 2.5e+80) || (!(z <= 5.3e+132) && (z <= 3e+140))))) {
		tmp = x * 0.5;
	} else {
		tmp = z * -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 ((z <= 550000.0d0) .or. (.not. (z <= 6.5d+36)) .and. (z <= 2.5d+80) .or. (.not. (z <= 5.3d+132)) .and. (z <= 3d+140)) then
        tmp = x * 0.5d0
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 550000.0) || (!(z <= 6.5e+36) && ((z <= 2.5e+80) || (!(z <= 5.3e+132) && (z <= 3e+140))))) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 550000.0) or (not (z <= 6.5e+36) and ((z <= 2.5e+80) or (not (z <= 5.3e+132) and (z <= 3e+140)))):
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 550000.0) || (!(z <= 6.5e+36) && ((z <= 2.5e+80) || (!(z <= 5.3e+132) && (z <= 3e+140)))))
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 550000.0) || (~((z <= 6.5e+36)) && ((z <= 2.5e+80) || (~((z <= 5.3e+132)) && (z <= 3e+140)))))
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 550000.0], And[N[Not[LessEqual[z, 6.5e+36]], $MachinePrecision], Or[LessEqual[z, 2.5e+80], And[N[Not[LessEqual[z, 5.3e+132]], $MachinePrecision], LessEqual[z, 3e+140]]]]], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.5e5 or 6.4999999999999998e36 < z < 2.4999999999999998e80 or 5.3e132 < z < 2.99999999999999997e140

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

   2. Taylor expanded in x around inf 57.7%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 5.5e5 < z < 6.4999999999999998e36 or 2.4999999999999998e80 < z < 5.3e132 or 2.99999999999999997e140 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 80.5%

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

      1. associate-*r* 80.5%

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

      2. mul-1-neg 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 550000 \lor \neg \left(z \leq 6.5 \cdot 10^{+36}\right) \land \left(z \leq 2.5 \cdot 10^{+80} \lor \neg \left(z \leq 5.3 \cdot 10^{+132}\right) \land z \leq 3 \cdot 10^{+140}\right):\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 7: 59.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 370000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;y - z \cdot y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+80} \lor \neg \left(z \leq 5.8 \cdot 10^{+132}\right) \land z \leq 3 \cdot 10^{+140}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 370000.0)
   (* x 0.5)
   (if (<= z 4.2e+36)
     (- y (* z y))
     (if (or (<= z 2.3e+80) (and (not (<= z 5.8e+132)) (<= z 3e+140)))
       (* x 0.5)
       (* z (- y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 370000.0) {
		tmp = x * 0.5;
	} else if (z <= 4.2e+36) {
		tmp = y - (z * y);
	} else if ((z <= 2.3e+80) || (!(z <= 5.8e+132) && (z <= 3e+140))) {
		tmp = x * 0.5;
	} else {
		tmp = z * -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 (z <= 370000.0d0) then
        tmp = x * 0.5d0
    else if (z <= 4.2d+36) then
        tmp = y - (z * y)
    else if ((z <= 2.3d+80) .or. (.not. (z <= 5.8d+132)) .and. (z <= 3d+140)) then
        tmp = x * 0.5d0
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 370000.0) {
		tmp = x * 0.5;
	} else if (z <= 4.2e+36) {
		tmp = y - (z * y);
	} else if ((z <= 2.3e+80) || (!(z <= 5.8e+132) && (z <= 3e+140))) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 370000.0:
		tmp = x * 0.5
	elif z <= 4.2e+36:
		tmp = y - (z * y)
	elif (z <= 2.3e+80) or (not (z <= 5.8e+132) and (z <= 3e+140)):
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 370000.0)
		tmp = Float64(x * 0.5);
	elseif (z <= 4.2e+36)
		tmp = Float64(y - Float64(z * y));
	elseif ((z <= 2.3e+80) || (!(z <= 5.8e+132) && (z <= 3e+140)))
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 370000.0)
		tmp = x * 0.5;
	elseif (z <= 4.2e+36)
		tmp = y - (z * y);
	elseif ((z <= 2.3e+80) || (~((z <= 5.8e+132)) && (z <= 3e+140)))
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 370000.0], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 4.2e+36], N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 2.3e+80], And[N[Not[LessEqual[z, 5.8e+132]], $MachinePrecision], LessEqual[z, 3e+140]]], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.7e5 or 4.20000000000000009e36 < z < 2.30000000000000004e80 or 5.7999999999999997e132 < z < 2.99999999999999997e140

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

   2. Taylor expanded in x around inf 57.7%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 3.7e5 < z < 4.20000000000000009e36

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-+l+ 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-rgt-identity 99.9%

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

      5. associate-+r+ 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around -inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. distribute-rgt-neg-in 80.1%

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

      3. sub-neg 80.1%

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

      4. mul-1-neg 80.1%

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

      5. sub-neg 80.1%

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

      6. +-commutative 80.1%

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

      7. distribute-neg-in 80.1%

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

      8. remove-double-neg 80.1%

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

      9. sub-neg 80.1%

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

      10. metadata-eval 80.1%

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

      11. +-commutative 80.1%

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

   6. Simplified 80.1%

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

   7. Taylor expanded in z around inf 70.9%

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

   8. Taylor expanded in y around 0 70.9%

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

      1. sub-neg 70.9%

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

      2. distribute-lft-out 70.9%

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

      3. *-rgt-identity 70.9%

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

      4. distribute-rgt-neg-out 70.9%

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

      5. *-commutative 70.9%

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

      6. unsub-neg 70.9%

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

      7. *-commutative 70.9%

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

   10. Simplified 70.9%

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

   if 2.30000000000000004e80 < z < 5.7999999999999997e132 or 2.99999999999999997e140 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 82.0%

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

      1. associate-*r* 82.0%

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

      2. mul-1-neg 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 370000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;y - z \cdot y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+80} \lor \neg \left(z \leq 5.8 \cdot 10^{+132}\right) \land z \leq 3 \cdot 10^{+140}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 8: 75.7% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 - z \cdot y \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return (x * 0.5) - (z * y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Step-by-step derivation

   1. add-cube-cbrt 99.2%

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

   2. pow3 99.2%

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

   3. sub-neg 99.2%

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

   4. associate-+l+ 99.2%

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

   5. +-commutative 99.2%

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

   6. sub-neg 99.2%

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

3. Applied egg-rr 99.2%

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

4. Taylor expanded in z around inf 76.9%

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

   1. associate-*r* 76.9%

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

   2. neg-mul-1 76.9%

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

   3. *-commutative 76.9%

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

6. Simplified 76.9%

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

   1. *-commutative 76.9%

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

   2. distribute-rgt-neg-out 76.9%

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

   3. unsub-neg 76.9%

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

   4. *-commutative 76.9%

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

8. Applied egg-rr 76.9%

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

9. Final simplification 76.9%

   \[\leadsto x \cdot 0.5 - z \cdot y \]

Alternative 9: 40.4% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 0.5))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return x * 0.5;
}

Python

def code(x, y, z):
	return x * 0.5

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

2. Taylor expanded in x around inf 42.4%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

3. Final simplification 42.4%

   \[\leadsto x \cdot 0.5 \]

Alternative 10: 1.8% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Step-by-step derivation

   1. sub-neg 99.9%

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

   2. associate-+l+ 99.9%

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

   3. distribute-lft-in 99.9%

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

   4. *-rgt-identity 99.9%

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

   5. associate-+r+ 99.9%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around -inf 58.9%

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

   1. mul-1-neg 58.9%

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

   2. distribute-rgt-neg-in 58.9%

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

   3. sub-neg 58.9%

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

   4. mul-1-neg 58.9%

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

   5. sub-neg 58.9%

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

   6. +-commutative 58.9%

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

   7. distribute-neg-in 58.9%

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

   8. remove-double-neg 58.9%

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

   9. sub-neg 58.9%

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

   10. metadata-eval 58.9%

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

   11. +-commutative 58.9%

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

6. Simplified 58.9%

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

7. Taylor expanded in z around inf 36.3%

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

8. Taylor expanded in z around 0 1.9%

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

9. Final simplification 1.9%

   \[\leadsto y \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023238 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))