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

Percentage Accurate: 99.9% → 99.9%

Time: 8.2s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] * y + 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. 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. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift--.f64 N/A

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

   9. associate-+r- N/A

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

   10. lower--.f64 N/A

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

   11. lower-+.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 73.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) + \log z\right) \cdot y\\ t_1 := \mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+278}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ (- 1.0 z) (log z)) y)) (t_1 (fma (- z) y (* x 0.5))))
   (if (<= t_0 -1e+278) t_1 (if (<= t_0 -2e+172) (fma (log z) y y) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 - z) + log(z)) * y;
	double t_1 = fma(-z, y, (x * 0.5));
	double tmp;
	if (t_0 <= -1e+278) {
		tmp = t_1;
	} else if (t_0 <= -2e+172) {
		tmp = fma(log(z), y, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 - z) + log(z)) * y)
	t_1 = fma(Float64(-z), y, Float64(x * 0.5))
	tmp = 0.0
	if (t_0 <= -1e+278)
		tmp = t_1;
	elseif (t_0 <= -2e+172)
		tmp = fma(log(z), y, y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+278], t$95$1, If[LessEqual[t$95$0, -2e+172], N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < -9.99999999999999964e277 or -2.0000000000000002e172 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 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 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if -9.99999999999999964e277 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < -2.0000000000000002e172

   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 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - z\right) + \log z\right) \cdot y \leq -1 \cdot 10^{+278}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{elif}\;\left(\left(1 - z\right) + \log z\right) \cdot y \leq -2 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 400000:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (log z) z) y y)))
   (if (<= y -3.2e+248) t_0 (if (<= y 400000.0) (fma (- z) y (* x 0.5)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((log(z) - z), y, y);
	double tmp;
	if (y <= -3.2e+248) {
		tmp = t_0;
	} else if (y <= 400000.0) {
		tmp = fma(-z, y, (x * 0.5));
	} 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 <= -3.2e+248)
		tmp = t_0;
	elseif (y <= 400000.0)
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	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, -3.2e+248], t$95$0, If[LessEqual[y, 400000.0], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999985e248 or 4e5 < 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 90.8

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

   5. Applied rewrites 90.8%

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

   if -3.19999999999999985e248 < y < 4e5

   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 84.8

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

   5. Applied rewrites 84.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.8

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

   7. Applied rewrites 84.8%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.042:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.042) (fma 0.5 x (fma (log z) y y)) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.042) {
		tmp = fma(0.5, x, fma(log(z), y, y));
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.042)
		tmp = fma(0.5, x, fma(log(z), y, y));
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.042], N[(0.5 * x + N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0420000000000000026

   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.0420000000000000026 < 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 98.6

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

   5. Applied rewrites 98.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5 + N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + 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. 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 99.8

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

   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 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+r+ N/A

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

   4. distribute-rgt-in N/A

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

   5. lift--.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. sub-neg N/A

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

   13. neg-mul-1 N/A

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

   14. lift-log.f64 N/A

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

   15. associate-+l+ N/A

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

   16. lift-fma.f64 N/A

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

   17. +-commutative N/A

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

   18. distribute-lft1-in N/A

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

   19. lift-*.f64 N/A

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

   20. associate-+r+ N/A

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

6. Applied rewrites 99.8%

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

Alternative 6: 60.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 3.6e+49) (* 0.5 x) (* (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.6e+49) {
		tmp = 0.5 * x;
	} 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 <= 3.6d+49) then
        tmp = 0.5d0 * x
    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 <= 3.6e+49) {
		tmp = 0.5 * x;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.6e+49:
		tmp = 0.5 * x
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.6e+49)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(Float64(-z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.6e+49)
		tmp = 0.5 * x;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.6e+49], N[(0.5 * x), $MachinePrecision], N[((-z) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.59999999999999996e49

   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 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 55.2%

       \[\leadsto 0.5 \cdot x \]

   if 3.59999999999999996e49 < 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. *-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 70.6

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

   5. Applied rewrites 70.6%

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

Alternative 7: 75.0% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[((-z) * y + 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. 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 73.8

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

5. Applied rewrites 73.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 73.8

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

7. Applied rewrites 73.8%

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

Alternative 8: 41.1% 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. 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 71.2

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

5. Applied rewrites 71.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 65.1%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 46.1%

    \[\leadsto 0.5 \cdot x \]
12. 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 2024327 
(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)))))