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

Percentage Accurate: 99.9% → 99.6%

Time: 11.2s

Alternatives: 10

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 2: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log z - z\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+126}:\\ \;\;\;\;t\_0 \cdot y + y\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log z) z)))
   (if (<= y -2.1e+126)
     (+ (* t_0 y) y)
     (if (<= y 3.5e+58) (fma (- z) y (* x 0.5)) (fma t_0 y y)))))

C

double code(double x, double y, double z) {
	double t_0 = log(z) - z;
	double tmp;
	if (y <= -2.1e+126) {
		tmp = (t_0 * y) + y;
	} else if (y <= 3.5e+58) {
		tmp = fma(-z, y, (x * 0.5));
	} else {
		tmp = fma(t_0, y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(z) - z)
	tmp = 0.0
	if (y <= -2.1e+126)
		tmp = Float64(Float64(t_0 * y) + y);
	elseif (y <= 3.5e+58)
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	else
		tmp = fma(t_0, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, -2.1e+126], N[(N[(t$95$0 * y), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[y, 3.5e+58], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * y + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999999e126

   1. Initial program 99.7%

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

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]
   6. 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 95.0

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 95.1%

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

   if -2.0999999999999999e126 < y < 3.4999999999999997e58

   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 83.9

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

   5. Applied rewrites 83.9%

      \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(-z\right)} \]
   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 83.9

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

   7. Applied rewrites 83.9%

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

   if 3.4999999999999997e58 < y

   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 y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]
   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 86.4

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

   5. Applied rewrites 86.4%

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

Alternative 3: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (log z) z) y y)))
   (if (<= y -1.1e+126) t_0 (if (<= y 3.5e+58) (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 <= -1.1e+126) {
		tmp = t_0;
	} else if (y <= 3.5e+58) {
		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 <= -1.1e+126)
		tmp = t_0;
	elseif (y <= 3.5e+58)
		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, -1.1e+126], t$95$0, If[LessEqual[y, 3.5e+58], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.09999999999999999e126 or 3.4999999999999997e58 < y

   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 y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]
   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 90.1

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

   5. Applied rewrites 90.1%

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

   if -1.09999999999999999e126 < y < 3.4999999999999997e58

   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 83.9

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

   5. Applied rewrites 83.9%

      \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(-z\right)} \]
   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 83.9

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

   7. Applied rewrites 83.9%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.55:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.55) (fma x 0.5 (fma (log z) y y)) (fma (- 1.0 z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.55) {
		tmp = fma(x, 0.5, fma(log(z), y, y));
	} else {
		tmp = fma((1.0 - z), y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.55)
		tmp = fma(x, 0.5, fma(log(z), y, y));
	else
		tmp = fma(Float64(1.0 - z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.55], N[(x * 0.5 + N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.55000000000000004

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.55000000000000004 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(1 - z, y, \color{blue}{x \cdot 0.5}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[Log[z], $MachinePrecision] + 1.0), $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%

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

Alternative 6: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\log z \cdot y + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.2e-103) (+ (* (log z) y) y) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.2e-103) {
		tmp = (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 <= 5.2e-103)
		tmp = Float64(Float64(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, 5.2e-103], N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] + y), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.19999999999999993e-103

   1. Initial program 99.7%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   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. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]
   6. 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 53.4

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\log z, y, y\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 53.4%

       \[\leadsto \mathsf{fma}\left(\log z, y, y\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 53.5%

       \[\leadsto \log z \cdot y + \color{blue}{y} \]

   if 5.19999999999999993e-103 < 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 91.8

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

   5. Applied rewrites 91.8%

      \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(-z\right)} \]
   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 91.8

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

   7. Applied rewrites 91.8%

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

Alternative 7: 74.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.2e-103) (fma (log z) y y) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.2e-103) {
		tmp = 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 <= 5.2e-103)
		tmp = 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, 5.2e-103], N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.19999999999999993e-103

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   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 53.4%

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

   if 5.19999999999999993e-103 < 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 91.8

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

   5. Applied rewrites 91.8%

      \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(-z\right)} \]
   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 91.8

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

   7. Applied rewrites 91.8%

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

Alternative 8: 60.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.3 \cdot 10^{+40}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 6.3e+40) (* x 0.5) (* (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.3e+40) {
		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 <= 6.3d+40) 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 <= 6.3e+40) {
		tmp = x * 0.5;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 6.3e+40:
		tmp = x * 0.5
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 6.3e+40)
		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 <= 6.3e+40)
		tmp = x * 0.5;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 6.3e+40], N[(x * 0.5), $MachinePrecision], N[((-z) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.30000000000000006e40

   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 y around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} \]

      2. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if 6.30000000000000006e40 < 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 85.6

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

   5. Applied rewrites 85.6%

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

Alternative 9: 74.4% 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 76.4

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

5. Applied rewrites 76.4%

   \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(-z\right)} \]
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 76.4

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

7. Applied rewrites 76.4%

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

Alternative 10: 39.7% 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 y around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2}} \]

   2. lower-*.f64 36.1

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

5. Applied rewrites 36.1%

   \[\leadsto \color{blue}{x \cdot 0.5} \]
6. 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 2024277 
(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)))))