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

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z - \left(z - 1\right) \leq -100000:\\ \;\;\;\;\left(-z\right) \cdot y + 0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (log z) (- z 1.0)) -100000.0)
   (+ (* (- z) y) (* 0.5 x))
   (fma 0.5 x (fma (log z) y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(z) - (z - 1.0)) <= -100000.0) {
		tmp = (-z * y) + (0.5 * x);
	} else {
		tmp = fma(0.5, x, fma(log(z), y, y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(z) - Float64(z - 1.0)) <= -100000.0)
		tmp = Float64(Float64(Float64(-z) * y) + Float64(0.5 * x));
	else
		tmp = fma(0.5, x, fma(log(z), y, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[z], $MachinePrecision] - N[(z - 1.0), $MachinePrecision]), $MachinePrecision], -100000.0], N[(N[((-z) * y), $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], N[(0.5 * x + 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)) < -1e5

   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 99.2

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

   5. Applied rewrites 99.2%

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

   if -1e5 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))

   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 z around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z - \left(z - 1\right) \leq -100000:\\ \;\;\;\;\left(-z\right) \cdot y + 0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+84}:\\ \;\;\;\;\left(-z\right) \cdot y + 0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (log z) z) y y)))
   (if (<= y -4.5e+87) t_0 (if (<= y 1.9e+84) (+ (* (- z) y) (* 0.5 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((log(z) - z), y, y);
	double tmp;
	if (y <= -4.5e+87) {
		tmp = t_0;
	} else if (y <= 1.9e+84) {
		tmp = (-z * y) + (0.5 * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(log(z) - z), y, y)
	tmp = 0.0
	if (y <= -4.5e+87)
		tmp = t_0;
	elseif (y <= 1.9e+84)
		tmp = Float64(Float64(Float64(-z) * y) + Float64(0.5 * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + y), $MachinePrecision]}, If[LessEqual[y, -4.5e+87], t$95$0, If[LessEqual[y, 1.9e+84], N[(N[((-z) * y), $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5000000000000003e87 or 1.9e84 < y

   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 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-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if -4.5000000000000003e87 < y < 1.9e84

   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 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+84}:\\ \;\;\;\;\left(-z\right) \cdot y + 0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z - \left(z - 1\right) \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (- (log z) (- z 1.0)) -5e+22) (* (- z) y) (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(z) - (z - 1.0)) <= -5e+22) {
		tmp = -z * y;
	} else {
		tmp = 0.5 * x;
	}
	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 ((log(z) - (z - 1.0d0)) <= (-5d+22)) then
        tmp = -z * y
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((Math.log(z) - (z - 1.0)) <= -5e+22) {
		tmp = -z * y;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.log(z) - (z - 1.0)) <= -5e+22:
		tmp = -z * y
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(z) - Float64(z - 1.0)) <= -5e+22)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((log(z) - (z - 1.0)) <= -5e+22)
		tmp = -z * y;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[z], $MachinePrecision] - N[(z - 1.0), $MachinePrecision]), $MachinePrecision], -5e+22], N[((-z) * y), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -4.9999999999999996e22

   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if -4.9999999999999996e22 < (+.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. +-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. distribute-rgt-in N/A

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

      6. associate-+l+ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-fma.f64 99.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 49.5%

       \[\leadsto 0.5 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z - \left(z - 1\right) \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y + 0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.25e-161) (fma (log z) y y) (+ (* (- z) y) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.25e-161) {
		tmp = fma(log(z), y, y);
	} else {
		tmp = (-z * y) + (0.5 * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.25e-161)
		tmp = fma(log(z), y, y);
	else
		tmp = Float64(Float64(Float64(-z) * y) + Float64(0.5 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.25e-161], N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision], N[(N[((-z) * y), $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.25e-161

   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. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.0%

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

   if 1.25e-161 < z

   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 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y + 0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[((-z) * y), $MachinePrecision] + N[(0.5 * x), $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 76.3

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

5. Applied rewrites 76.3%

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

6. Final simplification 76.3%

   \[\leadsto \left(-z\right) \cdot y + 0.5 \cdot x \]
7. Add Preprocessing

Alternative 7: 39.8% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * x), $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. distribute-rgt-in N/A

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

   6. associate-+l+ N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-fma.f64 99.9

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 84.0%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 36.1%

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

Alternative 8: 1.8% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ z \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z * y), $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

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 42.6

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

5. Applied rewrites 42.6%

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

7. Applied rewrites 33.3%

   \[\leadsto \frac{0 - z \cdot z}{0 + z} \cdot y \]
8. Step-by-step derivation

9. Applied rewrites 1.8%

   \[\leadsto \color{blue}{z \cdot y} \]
10. 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 2024298 
(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)))))