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

Percentage Accurate: 99.9% → 99.9%

Time: 10.0s

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. distribute-rgt-in N/A

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

   7. associate-+l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-fma.f64 99.9

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- z) y (* x 0.5))))
   (if (<= (* x 0.5) -4e-97)
     t_0
     (if (<= (* x 0.5) 2e-91) (fma (- (log z) z) y y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-z, y, (x * 0.5));
	double tmp;
	if ((x * 0.5) <= -4e-97) {
		tmp = t_0;
	} else if ((x * 0.5) <= 2e-91) {
		tmp = fma((log(z) - z), y, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-z), y, Float64(x * 0.5))
	tmp = 0.0
	if (Float64(x * 0.5) <= -4e-97)
		tmp = t_0;
	elseif (Float64(x * 0.5) <= 2e-91)
		tmp = fma(Float64(log(z) - z), y, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -4e-97], t$95$0, If[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-91], N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -4.00000000000000014e-97 or 2.00000000000000004e-91 < (*.f64 x #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.1

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

   7. Applied rewrites 84.1%

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

   if -4.00000000000000014e-97 < (*.f64 x #s(literal 1/2 binary64)) < 2.00000000000000004e-91

   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 84.9

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

   5. Applied rewrites 84.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0074:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.0074) (fma x 0.5 (fma (log z) y y)) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.0074) {
		tmp = fma(x, 0.5, 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 <= 0.0074)
		tmp = fma(x, 0.5, 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, 0.0074], N[(x * 0.5 + 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 < 0.0074000000000000003

   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 0.0074000000000000003 < 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. *-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 100.0

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

      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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.8

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

   7. Applied rewrites 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0074:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. distribute-rgt-in N/A

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

   7. associate-+l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-fma.f64 99.9

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. associate--l+ N/A

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

   12. lift-+.f64 N/A

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

   13. lift--.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. associate--l+ N/A

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

   20. lift--.f64 N/A

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

   21. +-commutative N/A

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

   22. distribute-lft1-in N/A

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

   23. lift-fma.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.9

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.9

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. distribute-lft-out-- N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. associate-+r+ N/A

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

   10. mul-1-neg N/A

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

   11. +-commutative N/A

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

   12. distribute-rgt-in N/A

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

   13. *-lft-identity N/A

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

   14. lower-fma.f64 N/A

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

   15. mul-1-neg N/A

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

   16. sub-neg N/A

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

   17. lower--.f64 N/A

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

   18. lower-log.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 6: 74.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+268}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.4e+268) (fma (- z) y (* x 0.5)) (fma (log z) y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+268) {
		tmp = fma(-z, y, (x * 0.5));
	} else {
		tmp = fma(log(z), y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.4e+268)
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	else
		tmp = fma(log(z), y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.4e+268], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.3999999999999998e268

   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. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if 6.3999999999999998e268 < y

   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 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+268}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2050000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2050000000.0) (* x 0.5) (fma (- z) y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2050000000.0) {
		tmp = x * 0.5;
	} else {
		tmp = fma(-z, y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2050000000.0)
		tmp = Float64(x * 0.5);
	else
		tmp = fma(Float64(-z), y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2050000000.0], N[(x * 0.5), $MachinePrecision], N[((-z) * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e9

   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 55.6

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

   5. Applied rewrites 55.6%

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

   if 2.05e9 < 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 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 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 78.0%

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

Alternative 8: 61.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2050000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2050000000.0) (* x 0.5) (* (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2050000000.0) {
		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 <= 2050000000.0d0) 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 <= 2050000000.0) {
		tmp = x * 0.5;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2050000000.0:
		tmp = x * 0.5
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2050000000.0)
		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 <= 2050000000.0)
		tmp = x * 0.5;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2050000000.0], N[(x * 0.5), $MachinePrecision], N[((-z) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05e9

   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 55.6

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

   5. Applied rewrites 55.6%

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

   if 2.05e9 < 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 78.0

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

   5. Applied rewrites 78.0%

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

Alternative 9: 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. 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. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 75.7

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

7. Applied rewrites 75.7%

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

8. Final simplification 75.7%

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

Alternative 10: 40.3% 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 40.8

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

5. Applied rewrites 40.8%

   \[\leadsto \color{blue}{x \cdot 0.5} \]
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 2024244 
(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)))))