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

Percentage Accurate: 99.9% → 99.9%

Time: 11.5s

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-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -3 \cdot 10^{-64}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;x \cdot 0.5 \leq 0.0001:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -3e-64)
   (- (* x 0.5) (* y z))
   (if (<= (* x 0.5) 0.0001)
     (+ y (* y (- (log z) z)))
     (fma y (- z) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -3e-64) {
		tmp = (x * 0.5) - (y * z);
	} else if ((x * 0.5) <= 0.0001) {
		tmp = y + (y * (log(z) - z));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -3e-64)
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	elseif (Float64(x * 0.5) <= 0.0001)
		tmp = Float64(y + Float64(y * Float64(log(z) - z)));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -3e-64], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 0.0001], N[(y + N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x 1/2) < -3.0000000000000001e-64

   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. flip-+ 79.5%

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

      2. clear-num 79.4%

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

      3. un-div-inv 79.4%

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

      4. clear-num 79.4%

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

      5. flip-+ 99.9%

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

      6. sub-neg 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 92.2%

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

      1. associate-*r* 92.2%

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

      2. neg-mul-1 92.2%

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

   6. Simplified 92.2%

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

      1. cancel-sign-sub-inv 92.2%

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

      2. *-commutative 92.2%

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

   8. Applied egg-rr 92.2%

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

   if -3.0000000000000001e-64 < (*.f64 x 1/2) < 1.00000000000000005e-4

   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-rgt-in 99.7%

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

      4. *-lft-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. *-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-rgt-in 99.7%

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

      10. distribute-lft-neg-out 99.7%

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

      11. *-commutative 99.7%

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

      12. unsub-neg 99.7%

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

      13. *-commutative 99.7%

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

      14. distribute-lft-out-- 99.7%

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

   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 x around 0 90.6%

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

   if 1.00000000000000005e-4 < (*.f64 x 1/2)

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 88.9%

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

      1. mul-1-neg 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -3 \cdot 10^{-64}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;x \cdot 0.5 \leq 0.0001:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 0.00679999999999999962

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. 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-rgt-in 99.7%

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

      4. *-lft-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. *-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-rgt-in 99.7%

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

      10. distribute-lft-neg-out 99.7%

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

      11. *-commutative 99.7%

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

      12. unsub-neg 99.7%

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

      13. *-commutative 99.7%

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

      14. distribute-lft-out-- 99.7%

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

   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 98.3%

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

   if 0.00679999999999999962 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 98.6%

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

      1. mul-1-neg 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 98.4%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 0.00679999999999999962

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 98.4%

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

   if 0.00679999999999999962 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 98.6%

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

      1. mul-1-neg 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 98.5%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 6: 74.6% 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-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in z around inf 74.7%

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

   1. mul-1-neg 74.7%

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

6. Simplified 74.7%

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

7. Final simplification 74.7%

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

Alternative 7: 74.6% 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. flip-+ 86.0%

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

   2. clear-num 86.0%

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

   3. un-div-inv 85.9%

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

   4. clear-num 85.9%

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

   5. flip-+ 99.7%

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

   6. sub-neg 99.7%

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

   7. associate-+l+ 99.7%

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

   8. +-commutative 99.7%

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

   9. sub-neg 99.7%

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

3. Applied egg-rr 99.7%

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

4. Taylor expanded in z around inf 74.7%

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

   1. associate-*r* 74.7%

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

   2. neg-mul-1 74.7%

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

6. Simplified 74.7%

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

   1. cancel-sign-sub-inv 74.7%

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

   2. *-commutative 74.7%

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

8. Applied egg-rr 74.7%

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

9. Final simplification 74.7%

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

Alternative 8: 39.7% 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 38.8%

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

3. Final simplification 38.8%

   \[\leadsto x \cdot 0.5 \]

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 2023297 
(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)))))