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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -500:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{-79}:\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -500.0)
   (fma y (- z) (* x 0.5))
   (if (<= (* x 0.5) 1e-79)
     (* y (+ (- 1.0 z) (log z)))
     (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -500.0) {
		tmp = fma(y, -z, (x * 0.5));
	} else if ((x * 0.5) <= 1e-79) {
		tmp = y * ((1.0 - z) + log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -500.0)
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	elseif (Float64(x * 0.5) <= 1e-79)
		tmp = Float64(y * Float64(Float64(1.0 - z) + log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -500.0], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 1e-79], N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x 1/2) < -500

   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 87.9%

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

      1. mul-1-neg 87.9%

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

   6. Simplified 87.9%

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

   if -500 < (*.f64 x 1/2) < 1e-79

   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 99.0%

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

   3. Taylor expanded in x around 0 91.3%

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

      1. +-commutative 91.3%

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

      2. +-commutative 91.3%

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

      3. distribute-lft-in 91.3%

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

      4. *-rgt-identity 91.3%

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

      5. associate-*r* 91.3%

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

      6. neg-mul-1 91.3%

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

      7. *-commutative 91.3%

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

      8. associate-+l+ 91.3%

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

      9. *-un-lft-identity 91.3%

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

      10. distribute-rgt-neg-out 91.3%

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

      11. distribute-lft-neg-in 91.3%

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

      12. distribute-rgt-in 91.2%

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

      13. sub-neg 91.2%

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

      14. +-commutative 91.2%

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

      15. distribute-lft-out 92.1%

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

   5. Applied egg-rr 92.1%

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

   if 1e-79 < (*.f64 x 1/2)

   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 85.1%

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

      1. mul-1-neg 85.1%

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

      2. *-commutative 85.1%

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

      3. distribute-rgt-neg-in 85.1%

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

   4. Simplified 85.1%

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

      1. *-commutative 85.1%

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

      2. neg-mul-1 85.1%

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

      3. associate-*r* 85.1%

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

      4. mul-1-neg 85.1%

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

      5. unsub-neg 85.1%

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

   6. Applied egg-rr 85.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -500:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{-79}:\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 3: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ t_1 := x \cdot 0.5 - y \cdot z\\ \mathbf{if}\;z \leq 5.8 \cdot 10^{-274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))) (t_1 (- (* x 0.5) (* y z))))
   (if (<= z 5.8e-274)
     t_1
     (if (<= z 1.9e-127)
       t_0
       (if (<= z 8.5e-116)
         t_1
         (if (<= z 2.8e-72) t_0 (fma y (- z) (* x 0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double t_1 = (x * 0.5) - (y * z);
	double tmp;
	if (z <= 5.8e-274) {
		tmp = t_1;
	} else if (z <= 1.9e-127) {
		tmp = t_0;
	} else if (z <= 8.5e-116) {
		tmp = t_1;
	} else if (z <= 2.8e-72) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	t_1 = Float64(Float64(x * 0.5) - Float64(y * z))
	tmp = 0.0
	if (z <= 5.8e-274)
		tmp = t_1;
	elseif (z <= 1.9e-127)
		tmp = t_0;
	elseif (z <= 8.5e-116)
		tmp = t_1;
	elseif (z <= 2.8e-72)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.8e-274], t$95$1, If[LessEqual[z, 1.9e-127], t$95$0, If[LessEqual[z, 8.5e-116], t$95$1, If[LessEqual[z, 2.8e-72], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.79999999999999952e-274 or 1.90000000000000001e-127 < z < 8.4999999999999995e-116

   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 73.5%

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

      1. mul-1-neg 73.5%

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

      2. *-commutative 73.5%

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

      3. distribute-rgt-neg-in 73.5%

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

   4. Simplified 73.5%

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

      1. *-commutative 73.5%

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

      2. neg-mul-1 73.5%

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

      3. associate-*r* 73.5%

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

      4. mul-1-neg 73.5%

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

      5. unsub-neg 73.5%

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

   6. Applied egg-rr 73.5%

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

   if 5.79999999999999952e-274 < z < 1.90000000000000001e-127 or 8.4999999999999995e-116 < z < 2.7999999999999998e-72

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 99.7%

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

   3. Taylor expanded in x around 0 67.5%

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

   4. Taylor expanded in z around 0 67.5%

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

   if 2.7999999999999998e-72 < 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. 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 92.0%

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

      1. mul-1-neg 92.0%

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

   6. Simplified 92.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-274}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-127}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-116}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-273} \lor \neg \left(z \leq 2.6 \cdot 10^{-127} \lor \neg \left(z \leq 2.55 \cdot 10^{-113}\right) \land z \leq 6.2 \cdot 10^{-73}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 1.7e-273)
         (not
          (or (<= z 2.6e-127) (and (not (<= z 2.55e-113)) (<= z 6.2e-73)))))
   (- (* x 0.5) (* y z))
   (* y (+ 1.0 (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1.7e-273) || !((z <= 2.6e-127) || (!(z <= 2.55e-113) && (z <= 6.2e-73)))) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + log(z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= 1.7d-273) .or. (.not. (z <= 2.6d-127) .or. (.not. (z <= 2.55d-113)) .and. (z <= 6.2d-73))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * (1.0d0 + log(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1.7e-273) || !((z <= 2.6e-127) || (!(z <= 2.55e-113) && (z <= 6.2e-73)))) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 1.7e-273) or not ((z <= 2.6e-127) or (not (z <= 2.55e-113) and (z <= 6.2e-73))):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * (1.0 + math.log(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 1.7e-273) || !((z <= 2.6e-127) || (!(z <= 2.55e-113) && (z <= 6.2e-73))))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(1.0 + log(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 1.7e-273) || ~(((z <= 2.6e-127) || (~((z <= 2.55e-113)) && (z <= 6.2e-73)))))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * (1.0 + log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 1.7e-273], N[Not[Or[LessEqual[z, 2.6e-127], And[N[Not[LessEqual[z, 2.55e-113]], $MachinePrecision], LessEqual[z, 6.2e-73]]]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.69999999999999996e-273 or 2.59999999999999991e-127 < z < 2.54999999999999989e-113 or 6.19999999999999938e-73 < 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 89.8%

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

      1. mul-1-neg 89.8%

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

      2. *-commutative 89.8%

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

      3. distribute-rgt-neg-in 89.8%

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

   4. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. neg-mul-1 89.8%

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

      3. associate-*r* 89.8%

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

      4. mul-1-neg 89.8%

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

      5. unsub-neg 89.8%

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

   6. Applied egg-rr 89.8%

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

   if 1.69999999999999996e-273 < z < 2.59999999999999991e-127 or 2.54999999999999989e-113 < z < 6.19999999999999938e-73

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 99.7%

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

   3. Taylor expanded in x around 0 67.5%

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

   4. Taylor expanded in z around 0 67.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-273} \lor \neg \left(z \leq 2.6 \cdot 10^{-127} \lor \neg \left(z \leq 2.55 \cdot 10^{-113}\right) \land z \leq 6.2 \cdot 10^{-73}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.07:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.07) (+ (* 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 <= 0.07) {
		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 <= 0.07)
		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, 0.07], 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 < 0.070000000000000007

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 97.8%

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

   if 0.070000000000000007 < 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 99.5%

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

      1. mul-1-neg 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.07:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 7: 75.0% 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. Taylor expanded in z around inf 71.9%

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

   1. mul-1-neg 71.9%

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

   2. *-commutative 71.9%

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

   3. distribute-rgt-neg-in 71.9%

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

4. Simplified 71.9%

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

   1. *-commutative 71.9%

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

   2. neg-mul-1 71.9%

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

   3. associate-*r* 71.9%

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

   4. mul-1-neg 71.9%

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

   5. unsub-neg 71.9%

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

6. Applied egg-rr 71.9%

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

7. Final simplification 71.9%

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

Alternative 8: 60.7% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+48}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 5e+48) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5e+48) {
		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 <= 5d+48) 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 <= 5e+48) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5e+48:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5e+48)
		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 <= 5e+48)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5e+48], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.99999999999999973e48

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 44.1%

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

   if 4.99999999999999973e48 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   4. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

      3. associate-*r* 100.0%

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

      4. mul-1-neg 100.0%

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

      5. unsub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 73.8%

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

      1. associate-*r* 73.8%

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

      2. neg-mul-1 73.8%

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

   9. Simplified 73.8%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+48}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 9: 39.5% 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 36.2%

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

3. Final simplification 36.2%

   \[\leadsto x \cdot 0.5 \]

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 2023314 
(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)))))