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

Percentage Accurate: 99.9% → 99.9%

Time: 11.1s

Alternatives: 8

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   9. lower-fma.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - z\right) + \log z \leq -1000000000000:\\ \;\;\;\;\mathsf{fma}\left(-z, y, \mathsf{fma}\left(x, 0.5, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (- 1.0 z) (log z)) -1000000000000.0)
   (fma (- z) y (fma x 0.5 y))
   (fma x 0.5 (fma (log z) y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) + log(z)) <= -1000000000000.0) {
		tmp = fma(-z, y, fma(x, 0.5, y));
	} else {
		tmp = fma(x, 0.5, fma(log(z), y, y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(1.0 - z) + log(z)) <= -1000000000000.0)
		tmp = fma(Float64(-z), y, fma(x, 0.5, y));
	else
		tmp = fma(x, 0.5, fma(log(z), y, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], -1000000000000.0], N[((-z) * y + N[(x * 0.5 + y), $MachinePrecision]), $MachinePrecision], N[(x * 0.5 + N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -1e12

   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 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. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, y, \mathsf{fma}\left(x, \frac{1}{2}, y\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   if -1e12 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))

   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

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{\left(x \cdot \frac{1}{2}\right)}^{3} + {\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}^{3}}{\left(x \cdot \frac{1}{2}\right) \cdot \left(x \cdot \frac{1}{2}\right) + \left(\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right) \cdot \left(y \cdot \left(\left(1 - z\right) + \log z\right)\right) - \left(x \cdot \frac{1}{2}\right) \cdot \left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \frac{1}{2}\right) \cdot \left(x \cdot \frac{1}{2}\right) + \left(\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right) \cdot \left(y \cdot \left(\left(1 - z\right) + \log z\right)\right) - \left(x \cdot \frac{1}{2}\right) \cdot \left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)\right)}{{\left(x \cdot \frac{1}{2}\right)}^{3} + {\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot \frac{1}{2}\right) \cdot \left(x \cdot \frac{1}{2}\right) + \left(\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right) \cdot \left(y \cdot \left(\left(1 - z\right) + \log z\right)\right) - \left(x \cdot \frac{1}{2}\right) \cdot \left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)\right)}{{\left(x \cdot \frac{1}{2}\right)}^{3} + {\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}^{3}}}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot \frac{1}{2}\right)}^{3} + {\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)}^{3}}{\left(x \cdot \frac{1}{2}\right) \cdot \left(x \cdot \frac{1}{2}\right) + \left(\left(y \cdot \left(\left(1 - z\right) + \log z\right)\right) \cdot \left(y \cdot \left(\left(1 - z\right) + \log z\right)\right) - \left(x \cdot \frac{1}{2}\right) \cdot \left(y \cdot \left(\left(1 - z\right) + \log z\right)\right)\right)}}}} \]

      6. flip3-+ N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 96.8

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

   7. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(\log z, y, y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. 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 2024229 
(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)))))