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

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $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. 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. fma-def 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-234}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{-236}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -1e-234)
   (- (* x 0.5) (* y z))
   (if (<= (* x 0.5) 2e-236) (* y (+ 1.0 (log z))) (fma y (- z) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -1e-234) {
		tmp = (x * 0.5) - (y * z);
	} else if ((x * 0.5) <= 2e-236) {
		tmp = y * (1.0 + log(z));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -1e-234)
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	elseif (Float64(x * 0.5) <= 2e-236)
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-234], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-236], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x 1/2) < -9.9999999999999996e-235

   1. Initial program 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. +-commutative 99.8%

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

      3. add-cube-cbrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. fma-def 99.6%

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

      6. pow2 99.6%

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

      7. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around inf 75.6%

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

      1. associate-*r* 75.6%

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

      2. neg-mul-1 75.6%

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

   7. Simplified 75.6%

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

      1. distribute-lft-neg-out 75.6%

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

      2. unsub-neg 75.6%

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

   9. Applied egg-rr 75.6%

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

   if -9.9999999999999996e-235 < (*.f64 x 1/2) < 2.0000000000000001e-236

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 65.9%

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

   4. Taylor expanded in x around 0 65.9%

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

   if 2.0000000000000001e-236 < (*.f64 x 1/2)

   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. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 82.5%

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

      1. mul-1-neg 82.5%

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

   7. Simplified 82.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-z}, x \cdot 0.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-234}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{-236}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-234} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-236}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -1e-234) (not (<= (* x 0.5) 2e-236)))
   (- (* x 0.5) (* y z))
   (* y (+ 1.0 (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -1e-234) || !((x * 0.5) <= 2e-236)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + log(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) <= (-1d-234)) .or. (.not. ((x * 0.5d0) <= 2d-236))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * (1.0d0 + log(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -1e-234) || !((x * 0.5) <= 2e-236)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -1e-234) or not ((x * 0.5) <= 2e-236):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * (1.0 + math.log(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -1e-234) || !(Float64(x * 0.5) <= 2e-236))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(1.0 + log(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -1e-234) || ~(((x * 0.5) <= 2e-236)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * (1.0 + log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-234], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-236]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -9.9999999999999996e-235 or 2.0000000000000001e-236 < (*.f64 x 1/2)

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. +-commutative 99.9%

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

      3. add-cube-cbrt 99.6%

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

      4. associate-*l* 99.6%

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

      5. fma-def 99.6%

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

      6. pow2 99.6%

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

      7. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around inf 78.5%

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

      1. associate-*r* 78.5%

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

      2. neg-mul-1 78.5%

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

   7. Simplified 78.5%

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

      1. distribute-lft-neg-out 78.5%

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

      2. unsub-neg 78.5%

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

   9. Applied egg-rr 78.5%

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

   if -9.9999999999999996e-235 < (*.f64 x 1/2) < 2.0000000000000001e-236

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 65.9%

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

   4. Taylor expanded in x around 0 65.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-234} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-236}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0068:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.0068)
   (+ (* x 0.5) (* y (+ 1.0 (log z))))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.0068) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.0068)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.0068], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.00679999999999999962

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 99.6%

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

   if 0.00679999999999999962 < 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. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 97.8%

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

      1. mul-1-neg 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0068:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 74.8% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. distribute-rgt-in 99.4%

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

   2. +-commutative 99.4%

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

   3. add-cube-cbrt 99.1%

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

   4. associate-*l* 99.1%

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

   5. fma-def 99.5%

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

   6. pow2 99.5%

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

   7. *-commutative 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in z around inf 73.1%

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

   1. associate-*r* 73.1%

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

   2. neg-mul-1 73.1%

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

7. Simplified 73.1%

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

   1. distribute-lft-neg-out 73.1%

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

   2. unsub-neg 73.1%

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

9. Applied egg-rr 73.1%

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

10. Final simplification 73.1%

    \[\leadsto x \cdot 0.5 - y \cdot z \]
11. Add Preprocessing

Alternative 7: 40.1% 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. Add Preprocessing

3. Taylor expanded in x around inf 42.6%

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

4. Final simplification 42.6%

   \[\leadsto x \cdot 0.5 \]
5. Add Preprocessing

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 2024021 
(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)))))