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

Percentage Accurate: 99.9% → 99.9%

Time: 11.8s

Alternatives: 9

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-define 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. Add Preprocessing

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5000000000000:\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -5e-85)
   (fma y (- z) (* x 0.5))
   (if (<= (* x 0.5) 5000000000000.0)
     (* y (+ (- 1.0 z) (log z)))
     (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -5e-85) {
		tmp = fma(y, -z, (x * 0.5));
	} else if ((x * 0.5) <= 5000000000000.0) {
		tmp = y * ((1.0 - z) + log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -5e-85)
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	elseif (Float64(x * 0.5) <= 5000000000000.0)
		tmp = Float64(y * Float64(Float64(1.0 - z) + log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -5e-85], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 5000000000000.0], N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -5.0000000000000002e-85

   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-define 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. Taylor expanded in z around inf 84.4%

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

      1. neg-mul-1 84.4%

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

   7. Simplified 84.4%

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

   if -5.0000000000000002e-85 < (*.f64 x #s(literal 1/2 binary64)) < 5e12

   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. fma-define 99.8%

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

      3. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 66.3%

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

   6. Taylor expanded in x around 0 87.6%

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

      1. distribute-lft-out 87.7%

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

   8. Simplified 87.7%

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

   if 5e12 < (*.f64 x #s(literal 1/2 binary64))

   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-define 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 88.4%

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

      1. neg-mul-1 88.4%

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

   7. Simplified 88.4%

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

   8. Taylor expanded in y around 0 88.4%

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

      1. +-commutative 88.4%

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

      2. *-commutative 88.4%

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

      3. mul-1-neg 88.4%

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

      4. unsub-neg 88.4%

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

      5. *-commutative 88.4%

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

   10. Simplified 88.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5000000000000:\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-293}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.2e-293)
   (* x 0.5)
   (if (<= z 1.4e-69) (* y (+ 1.0 (log z))) (fma y (- z) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.2e-293) {
		tmp = x * 0.5;
	} else if (z <= 1.4e-69) {
		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 (z <= 5.2e-293)
		tmp = Float64(x * 0.5);
	elseif (z <= 1.4e-69)
		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[z, 5.2e-293], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 1.4e-69], 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 z < 5.1999999999999996e-293

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 67.9%

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

   if 5.1999999999999996e-293 < z < 1.3999999999999999e-69

   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 x around inf 89.9%

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

      1. associate-/l* 89.8%

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

      2. +-commutative 89.8%

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

      3. associate--l+ 89.8%

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

   5. Simplified 89.8%

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

   6. Taylor expanded in z around 0 89.8%

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

   7. Taylor expanded in x around 0 57.0%

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

      1. +-commutative 57.0%

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

   9. Simplified 57.0%

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

   if 1.3999999999999999e-69 < 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-define 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 92.8%

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

      1. neg-mul-1 92.8%

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

   7. Simplified 92.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-293}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{-293}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.6e-293)
   (* x 0.5)
   (if (<= z 1.45e-68) (* y (+ 1.0 (log z))) (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.6e-293) {
		tmp = x * 0.5;
	} else if (z <= 1.45e-68) {
		tmp = y * (1.0 + log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 5.6d-293) then
        tmp = x * 0.5d0
    else if (z <= 1.45d-68) then
        tmp = y * (1.0d0 + log(z))
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.6e-293) {
		tmp = x * 0.5;
	} else if (z <= 1.45e-68) {
		tmp = y * (1.0 + Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5.6e-293:
		tmp = x * 0.5
	elif z <= 1.45e-68:
		tmp = y * (1.0 + math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.6e-293)
		tmp = Float64(x * 0.5);
	elseif (z <= 1.45e-68)
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 5.6e-293)
		tmp = x * 0.5;
	elseif (z <= 1.45e-68)
		tmp = y * (1.0 + log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.6e-293], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 1.45e-68], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.60000000000000051e-293

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 67.9%

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

   if 5.60000000000000051e-293 < z < 1.45e-68

   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 x around inf 89.9%

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

      1. associate-/l* 89.8%

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

      2. +-commutative 89.8%

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

      3. associate--l+ 89.8%

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

   5. Simplified 89.8%

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

   6. Taylor expanded in z around 0 89.8%

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

   7. Taylor expanded in x around 0 57.0%

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

      1. +-commutative 57.0%

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

   9. Simplified 57.0%

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

   if 1.45e-68 < 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-define 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 92.8%

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

      1. neg-mul-1 92.8%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in y around 0 92.8%

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

      1. +-commutative 92.8%

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

      2. *-commutative 92.8%

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

      3. mul-1-neg 92.8%

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

      4. unsub-neg 92.8%

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

      5. *-commutative 92.8%

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

   10. Simplified 92.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{-293}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.2 \cdot 10^{-21}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 8.2e-21)
   (+ (* 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 <= 8.2e-21) {
		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 <= 8.2e-21)
		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, 8.2e-21], 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 < 8.19999999999999988e-21

   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 99.8%

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

   if 8.19999999999999988e-21 < 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-define 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 98.6%

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

      1. neg-mul-1 98.6%

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

   7. Simplified 98.6%

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

Alternative 6: 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. Add Preprocessing

Alternative 7: 60.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.4e+26) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e+26) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.4d+26) then
        tmp = x * 0.5d0
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e+26) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.4e+26:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.4e+26)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.4e+26)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.4e+26], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.40000000000000005e26

   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 x around inf 53.2%

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

   if 2.40000000000000005e26 < 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-define 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 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 74.0%

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

       1. neg-mul-1 74.0%

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

   13. Simplified 74.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% 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. 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-define 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. Taylor expanded in z around inf 75.0%

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

   1. neg-mul-1 75.0%

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

7. Simplified 75.0%

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

8. Taylor expanded in y around 0 75.0%

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

   1. +-commutative 75.0%

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

   2. *-commutative 75.0%

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

   3. mul-1-neg 75.0%

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

   4. unsub-neg 75.0%

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

   5. *-commutative 75.0%

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

10. Simplified 75.0%

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

11. Final simplification 75.0%

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

Alternative 9: 40.2% 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 42.2%

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

4. Final simplification 42.2%

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

Developer Target 1: 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 2024118 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

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