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

Percentage Accurate: 99.9% → 99.9%

Time: 12.6s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2} + y \cdot \left(\left(1 - z\right) + \log z\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot \frac{1}{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right)} + x \cdot \frac{1}{2} \]

   4. lift-+.f64 N/A

      \[\leadsto y \cdot \color{blue}{\left(\left(1 - z\right) + \log z\right)} + x \cdot \frac{1}{2} \]

   5. +-commutative N/A

      \[\leadsto y \cdot \color{blue}{\left(\log z + \left(1 - z\right)\right)} + x \cdot \frac{1}{2} \]

   6. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\log z \cdot y + \left(1 - z\right) \cdot y\right)} + x \cdot \frac{1}{2} \]

   7. associate-+l+ N/A

      \[\leadsto \color{blue}{\log z \cdot y + \left(\left(1 - z\right) \cdot y + x \cdot \frac{1}{2}\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log z, y, \left(1 - z\right) \cdot y + x \cdot \frac{1}{2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\log z, y, \color{blue}{y \cdot \left(1 - z\right)} + x \cdot \frac{1}{2}\right) \]

   10. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.5 + y \cdot \left(-z\right)\\ \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* x 0.5) (* y (- z)))))
   (if (<= (* x 0.5) -1e-134)
     t_0
     (if (<= (* x 0.5) 1e-52) (fma y (- (log z) z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 0.5) + (y * -z);
	double tmp;
	if ((x * 0.5) <= -1e-134) {
		tmp = t_0;
	} else if ((x * 0.5) <= 1e-52) {
		tmp = fma(y, (log(z) - z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 0.5) + Float64(y * Float64(-z)))
	tmp = 0.0
	if (Float64(x * 0.5) <= -1e-134)
		tmp = t_0;
	elseif (Float64(x * 0.5) <= 1e-52)
		tmp = fma(y, Float64(log(z) - z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 0.5), $MachinePrecision] + N[(y * (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-134], t$95$0, If[LessEqual[N[(x * 0.5), $MachinePrecision], 1e-52], N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -1.00000000000000004e-134 or 1e-52 < (*.f64 x #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(-1 \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      2. lower-neg.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if -1.00000000000000004e-134 < (*.f64 x #s(literal 1/2 binary64)) < 1e-52

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \log z + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}, y\right) \]

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.000135:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.000135) (fma x 0.5 (fma y (log z) y)) (+ (* x 0.5) (* y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.000135) {
		tmp = fma(x, 0.5, fma(y, log(z), y));
	} else {
		tmp = (x * 0.5) + (y * -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.000135)
		tmp = fma(x, 0.5, fma(y, log(z), y));
	else
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(-z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.000135], N[(x * 0.5 + N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] + N[(y * (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.35000000000000002e-4

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\left(1 - z\right) + \log z\right)} \]

      3. flip-+ N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\frac{\left(1 - z\right) \cdot \left(1 - z\right) - \log z \cdot \log z}{\left(1 - z\right) - \log z}} \]

      4. clear-num N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\frac{1}{\frac{\left(1 - z\right) - \log z}{\left(1 - z\right) \cdot \left(1 - z\right) - \log z \cdot \log z}}} \]

      5. un-div-inv N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\frac{y}{\frac{\left(1 - z\right) - \log z}{\left(1 - z\right) \cdot \left(1 - z\right) - \log z \cdot \log z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\frac{y}{\frac{\left(1 - z\right) - \log z}{\left(1 - z\right) \cdot \left(1 - z\right) - \log z \cdot \log z}}} \]

      7. clear-num N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\color{blue}{\frac{1}{\frac{\left(1 - z\right) \cdot \left(1 - z\right) - \log z \cdot \log z}{\left(1 - z\right) - \log z}}}} \]

      8. flip-+ N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\color{blue}{\left(1 - z\right) + \log z}}} \]

      9. lift-+.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\color{blue}{\left(1 - z\right) + \log z}}} \]

      10. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\color{blue}{z \cdot \left(\left(-1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z} + \frac{1}{z}\right) - 1\right)}}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{z \cdot \color{blue}{\left(\left(-1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z} + \frac{1}{z}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}}} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{z \cdot \left(\left(-1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z} + \frac{1}{z}\right) + \color{blue}{-1}\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\color{blue}{z \cdot \left(-1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z} + \frac{1}{z}\right) + z \cdot -1}}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{z \cdot \left(-1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z} + \frac{1}{z}\right) + \color{blue}{-1 \cdot z}}} \]

      5. mul-1-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{z \cdot \left(-1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z} + \frac{1}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\color{blue}{\mathsf{fma}\left(z, -1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z} + \frac{1}{z}, \mathsf{neg}\left(z\right)\right)}}} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \color{blue}{\frac{1}{z} + -1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z}}, \mathsf{neg}\left(z\right)\right)}} \]

      8. lower-+.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \color{blue}{\frac{1}{z} + -1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z}}, \mathsf{neg}\left(z\right)\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \color{blue}{\frac{1}{z}} + -1 \cdot \frac{\log \left(\frac{1}{z}\right)}{z}, \mathsf{neg}\left(z\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \frac{1}{z} + \color{blue}{\left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{z}\right)}{z}\right)\right)}, \mathsf{neg}\left(z\right)\right)}} \]

      11. distribute-frac-neg N/A

          \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \frac{1}{z} + \color{blue}{\frac{\mathsf{neg}\left(\log \left(\frac{1}{z}\right)\right)}{z}}, \mathsf{neg}\left(z\right)\right)}} \]

      12. log-rec N/A

          \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \frac{1}{z} + \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log z\right)\right)}\right)}{z}, \mathsf{neg}\left(z\right)\right)}} \]

      13. remove-double-neg N/A

          \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \frac{1}{z} + \frac{\color{blue}{\log z}}{z}, \mathsf{neg}\left(z\right)\right)}} \]

      14. lower-/.f64 N/A

          \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \frac{1}{z} + \color{blue}{\frac{\log z}{z}}, \mathsf{neg}\left(z\right)\right)}} \]

      15. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{1}{2} + \frac{y}{\frac{1}{\mathsf{fma}\left(z, \frac{1}{z} + \frac{\color{blue}{\log z}}{z}, \mathsf{neg}\left(z\right)\right)}} \]

      16. lower-neg.f64 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in z around 0

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(1 + \log z\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\log z + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\left(\log z \cdot y + 1 \cdot y\right)} \]

      3. *-lft-identity N/A

         \[\leadsto x \cdot \frac{1}{2} + \left(\log z \cdot y + \color{blue}{y}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\mathsf{fma}\left(\log z, y, y\right)} \]

      5. lower-log.f64 99.6

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

   10. Applied rewrites 99.6%

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

       1. lift-+.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{2} + \mathsf{fma}\left(\log z, y, y\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + \mathsf{fma}\left(\log z, y, y\right) \]

       3. lower-fma.f64 99.6

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

   12. Applied rewrites 99.6%

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

   if 1.35000000000000002e-4 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(-1 \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      2. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.000135:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, \mathsf{fma}\left(0.5, x, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.000135) (fma y (log z) (fma 0.5 x y)) (+ (* x 0.5) (* y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.000135) {
		tmp = fma(y, log(z), fma(0.5, x, y));
	} else {
		tmp = (x * 0.5) + (y * -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.000135)
		tmp = fma(y, log(z), fma(0.5, x, y));
	else
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(-z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.000135], N[(y * N[Log[z], $MachinePrecision] + N[(0.5 * x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] + N[(y * (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.35000000000000002e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + y \cdot \left(1 + \log z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1}{2} \cdot x + \color{blue}{\left(y \cdot 1 + y \cdot \log z\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot x + \left(\color{blue}{y} + y \cdot \log z\right) \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + y\right) + y \cdot \log z} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \log z + \left(\frac{1}{2} \cdot x + y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z, \frac{1}{2} \cdot x + y\right)} \]

      6. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log z}, \frac{1}{2} \cdot x + y\right) \]

      7. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1.35000000000000002e-4 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(-1 \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      2. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x + y \cdot \left(\left(1 + \log z\right) - z\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{1}{2} \cdot x + y \cdot \color{blue}{\left(\left(1 + \log z\right) + \left(\mathsf{neg}\left(z\right)\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \frac{1}{2} \cdot x + y \cdot \color{blue}{\left(1 + \left(\log z + \left(\mathsf{neg}\left(z\right)\right)\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{1}{2} \cdot x + y \cdot \left(1 + \left(\log z + \color{blue}{-1 \cdot z}\right)\right) \]

   4. distribute-lft-in N/A

      \[\leadsto \frac{1}{2} \cdot x + \color{blue}{\left(y \cdot 1 + y \cdot \left(\log z + -1 \cdot z\right)\right)} \]

   5. *-rgt-identity N/A

      \[\leadsto \frac{1}{2} \cdot x + \left(\color{blue}{y} + y \cdot \left(\log z + -1 \cdot z\right)\right) \]

   6. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + y\right) + y \cdot \left(\log z + -1 \cdot z\right)} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{y \cdot \left(\log z + -1 \cdot z\right) + \left(\frac{1}{2} \cdot x + y\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z + -1 \cdot z, \frac{1}{2} \cdot x + y\right)} \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(y, \log z + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}, \frac{1}{2} \cdot x + y\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log z - z}, \frac{1}{2} \cdot x + y\right) \]

   11. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log z - z}, \frac{1}{2} \cdot x + y\right) \]

   12. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log z} - z, \frac{1}{2} \cdot x + y\right) \]

   13. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 6: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.8e-12) (fma y (log z) y) (+ (* x 0.5) (* y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.8e-12) {
		tmp = fma(y, log(z), y);
	} else {
		tmp = (x * 0.5) + (y * -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.8e-12)
		tmp = fma(y, log(z), y);
	else
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(-z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.8e-12], N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] + N[(y * (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8000000000000003e-12

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + y \cdot \left(1 + \log z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1}{2} \cdot x + \color{blue}{\left(y \cdot 1 + y \cdot \log z\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot x + \left(\color{blue}{y} + y \cdot \log z\right) \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + y\right) + y \cdot \log z} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \log z + \left(\frac{1}{2} \cdot x + y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z, \frac{1}{2} \cdot x + y\right)} \]

      6. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log z}, \frac{1}{2} \cdot x + y\right) \]

      7. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.2%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log z}, y\right) \]

   if 5.8000000000000003e-12 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(-1 \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      2. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 7: 75.4% accurate, 7.5× speedup

Math

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

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 * Float64(-z)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf

   \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(-1 \cdot z\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

   2. lower-neg.f64 70.9

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

5. Applied rewrites 70.9%

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

Alternative 8: 58.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{+86}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 3.8e+86) (* x 0.5) (- (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.8e+86) {
		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 <= 3.8d+86) 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 <= 3.8e+86) {
		tmp = x * 0.5;
	} else {
		tmp = -(z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.8e+86:
		tmp = x * 0.5
	else:
		tmp = -(z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.8e+86)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(-Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.8e+86)
		tmp = x * 0.5;
	else
		tmp = -(z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.8e+86], N[(x * 0.5), $MachinePrecision], (-N[(z * y), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < 3.79999999999999978e86

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 45.1

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

   5. Applied rewrites 45.1%

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

   if 3.79999999999999978e86 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{+86}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.4% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return x * 0.5;
}

Python

def code(x, y, z):
	return x * 0.5

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = x * 0.5;
end

Wolfram

code[x_, y_, z_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 37.5

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

5. Applied rewrites 37.5%

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

6. Final simplification 37.5%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

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