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

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

Alternatives: 8

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: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-126}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \left(\left(1 + \log z\right) \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.2e-126)
   (* x (- 0.5 (* z (/ y x))))
   (if (<= z 4.6e-50)
     (* x (* (+ 1.0 (log z)) (/ y x)))
     (fma y (- z) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.2e-126) {
		tmp = x * (0.5 - (z * (y / x)));
	} else if (z <= 4.6e-50) {
		tmp = x * ((1.0 + log(z)) * (y / x));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.2e-126)
		tmp = Float64(x * Float64(0.5 - Float64(z * Float64(y / x))));
	elseif (z <= 4.6e-50)
		tmp = Float64(x * Float64(Float64(1.0 + log(z)) * Float64(y / x)));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.2e-126], N[(x * N[(0.5 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e-50], N[(x * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.19999999999999998e-126

   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 inf 55.2%

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

      1. associate-*r* 55.2%

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

      2. neg-mul-1 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in x around inf 56.4%

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

      1. mul-1-neg 56.4%

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

      2. unsub-neg 56.4%

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

      3. *-commutative 56.4%

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

      4. associate-*r/ 56.8%

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

   8. Simplified 56.8%

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

   if 5.19999999999999998e-126 < z < 4.60000000000000039e-50

   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 91.1%

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

      1. mul-1-neg 91.1%

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

      2. *-commutative 91.1%

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

      3. distribute-rgt-neg-in 91.1%

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

      4. sub-neg 91.1%

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

      5. metadata-eval 91.1%

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

      6. +-commutative 91.1%

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

      7. mul-1-neg 91.1%

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

      8. unsub-neg 91.1%

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

      9. *-commutative 91.1%

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

      10. +-commutative 91.1%

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

      11. associate--l+ 91.1%

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

      12. +-commutative 91.1%

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

      13. associate-/l* 91.0%

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

      14. +-commutative 91.0%

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

   5. Simplified 91.0%

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

   6. Taylor expanded in z around 0 91.0%

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

   7. Taylor expanded in z around 0 91.1%

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

      1. distribute-lft-in 91.1%

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

      2. associate-/l* 90.9%

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

      3. +-commutative 90.9%

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

      4. +-commutative 90.9%

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

      5. remove-double-neg 90.9%

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

      6. log-rec 90.9%

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

      7. mul-1-neg 90.9%

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

      8. associate-/l* 91.0%

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

      9. distribute-lft-in 91.0%

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

   9. Simplified 91.0%

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

   10. Taylor expanded in y around inf 64.7%

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

       1. distribute-lft-in 64.2%

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

       2. associate-*r/ 64.2%

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

       3. *-rgt-identity 64.2%

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

       4. associate-*r/ 64.2%

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

       5. associate-*l/ 64.2%

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

       6. *-rgt-identity 64.2%

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

       7. distribute-lft-in 64.7%

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

       8. *-commutative 64.7%

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

   12. Simplified 64.7%

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

   if 4.60000000000000039e-50 < 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 92.9%

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

      1. neg-mul-1 92.9%

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

   7. Simplified 92.9%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.042:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.042) (+ (* 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 <= 0.042) {
		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 <= 0.042)
		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, 0.042], 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 < 0.0420000000000000026

   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 99.3%

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

      1. *-commutative 99.3%

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

   5. Simplified 99.3%

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

   if 0.0420000000000000026 < 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 98.7%

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

      1. neg-mul-1 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.042:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y * (-z) + 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. Taylor expanded in z around inf 75.8%

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

   1. neg-mul-1 75.8%

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

7. Simplified 75.8%

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

Alternative 6: 60.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{+18}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.95e+18) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.95e+18) {
		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 <= 1.95d+18) 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 <= 1.95e+18) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.95e+18:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.95e+18)
		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 <= 1.95e+18)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.95e+18], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.95e18

   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 inf 53.8%

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

      1. associate-*r* 53.8%

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

      2. neg-mul-1 53.8%

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

   5. Simplified 53.8%

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

   6. Taylor expanded in x around inf 51.9%

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

   if 1.95e18 < z

   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 z around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around -inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. sub-neg 87.6%

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

      3. associate-/l* 78.8%

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

      4. metadata-eval 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in x around 0 74.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{+18}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% 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. Taylor expanded in z around inf 75.8%

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

   1. associate-*r* 75.8%

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

   2. neg-mul-1 75.8%

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

5. Simplified 75.8%

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

   1. distribute-lft-neg-out 75.8%

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

   2. unsub-neg 75.8%

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

   3. add-sqr-sqrt 35.8%

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

   4. sqrt-unprod 48.8%

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

   5. sqr-neg 48.8%

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

   6. sqrt-unprod 17.1%

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

   7. add-sqr-sqrt 38.3%

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

   8. *-commutative 38.3%

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

   9. add-sqr-sqrt 17.1%

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

   10. sqrt-unprod 48.8%

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

   11. sqr-neg 48.8%

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

   12. sqrt-unprod 35.8%

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

   13. add-sqr-sqrt 75.8%

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

7. Applied egg-rr 75.8%

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

8. Final simplification 75.8%

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

Alternative 8: 40.6% 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 z around inf 75.8%

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

   1. associate-*r* 75.8%

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

   2. neg-mul-1 75.8%

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

5. Simplified 75.8%

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

6. Taylor expanded in x around inf 39.6%

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

7. Final simplification 39.6%

   \[\leadsto x \cdot 0.5 \]
8. 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 2024150 
(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)))))