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

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 11

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: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-28}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-50}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -1e-28)
   (- (* x 0.5) (* z y))
   (if (<= (* x 0.5) 5e-50)
     (* y (- (+ 1.0 (log z)) z))
     (fma (- z) y (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -1e-28) {
		tmp = (x * 0.5) - (z * y);
	} else if ((x * 0.5) <= 5e-50) {
		tmp = y * ((1.0 + log(z)) - z);
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -1e-28)
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	elseif (Float64(x * 0.5) <= 5e-50)
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-28], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 5e-50], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x 1/2) < -9.99999999999999971e-29

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. distribute-rgt-neg-out 95.2%

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

   4. Simplified 95.2%

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

   if -9.99999999999999971e-29 < (*.f64 x 1/2) < 4.99999999999999968e-50

   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. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-def 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around inf 91.4%

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

   if 4.99999999999999968e-50 < (*.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. *-commutative 99.9%

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

      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 86.7%

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

      1. neg-mul-1 86.7%

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

   6. Simplified 86.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-28}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-50}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.175:\\ \;\;\;\;\log z \cdot y + \left(y + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.175) (+ (* (log z) y) (+ y (* x 0.5))) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.175) {
		tmp = (log(z) * y) + (y + (x * 0.5));
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.175)
		tmp = Float64(Float64(log(z) * y) + Float64(y + Float64(x * 0.5)));
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.175], N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] + N[(y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.17499999999999999

   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 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 98.9%

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

   if 0.17499999999999999 < 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 98.7%

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

      1. neg-mul-1 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.175:\\ \;\;\;\;\log z \cdot y + \left(y + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(N[Log[z], $MachinePrecision] + N[(1.0 - 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. Final simplification 99.9%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-+r- 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 6: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.2e-131) (* y (+ 1.0 (log z))) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.2e-131) {
		tmp = y * (1.0 + log(z));
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.2e-131)
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.2e-131], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.2e-131

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. distribute-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

      5. associate-+r+ 99.6%

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

      6. fma-def 99.6%

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

      7. +-commutative 99.6%

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

      8. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 99.6%

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

   5. Taylor expanded in y around inf 58.1%

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

   if 3.2e-131 < z

   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. Taylor expanded in z around inf 88.8%

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

      1. neg-mul-1 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-130}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.05e-130) (* y (+ 1.0 (log z))) (- (* x 0.5) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.05e-130) {
		tmp = y * (1.0 + log(z));
	} 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 <= 1.05d-130) then
        tmp = y * (1.0d0 + log(z))
    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 <= 1.05e-130) {
		tmp = y * (1.0 + Math.log(z));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.05e-130:
		tmp = y * (1.0 + math.log(z))
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.05e-130)
		tmp = Float64(y * Float64(1.0 + log(z)));
	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 <= 1.05e-130)
		tmp = y * (1.0 + log(z));
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.05e-130], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.05000000000000001e-130

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. distribute-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

      5. associate-+r+ 99.6%

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

      6. fma-def 99.6%

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

      7. +-commutative 99.6%

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

      8. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 99.6%

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

   5. Taylor expanded in y around inf 58.1%

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

   if 1.05000000000000001e-130 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. distribute-rgt-neg-out 88.8%

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

   4. Simplified 88.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-130}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 8: 74.3% 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. Taylor expanded in z around inf 77.4%

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

   1. mul-1-neg 77.4%

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

   2. distribute-rgt-neg-out 77.4%

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

4. Simplified 77.4%

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

5. Final simplification 77.4%

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

Alternative 9: 58.9% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 9.8 \cdot 10^{+62}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 9.8e+62) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 9.8e+62) {
		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 <= 9.8d+62) 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 <= 9.8e+62) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 9.8e+62:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 9.8e+62)
		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 <= 9.8e+62)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 9.8e+62], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.7999999999999994e62

   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 51.1%

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

   if 9.7999999999999994e62 < 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 80.3%

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

      1. mul-1-neg 80.3%

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

      2. distribute-rgt-neg-out 80.3%

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

   6. Simplified 80.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.8 \cdot 10^{+62}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 10: 39.2% 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. Taylor expanded in x around inf 37.2%

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

3. Final simplification 37.2%

   \[\leadsto x \cdot 0.5 \]

Alternative 11: 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 99.8%

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

   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. Step-by-step derivation

   1. add-cube-cbrt 98.9%

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

   2. pow3 98.9%

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

5. Applied egg-rr 98.9%

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

6. Taylor expanded in y around inf 1.7%

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

7. Final simplification 1.7%

   \[\leadsto y \]

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 2023229 
(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)))))