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

Percentage Accurate: 99.9% → 99.9%

Time: 12.4s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. associate-+r+ N/A

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

   3. mul-1-neg N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. mul-1-neg N/A

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

   10. sub-neg N/A

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

   11. --lowering--.f64 N/A

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

   12. log-lowering-log.f64 N/A

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

   13. accelerator-lowering-fma.f64 99.9

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

5. Simplified 99.9%

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

Alternative 2: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{if}\;z \leq 2.4 \cdot 10^{-265}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-104}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\frac{y}{x}, -z, 0.5\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, 0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (log z) y)))
   (if (<= z 2.4e-265)
     t_0
     (if (<= z 6e-104)
       (* x (fma (/ y x) (- z) 0.5))
       (if (<= z 3.2e-65) t_0 (fma y (- z) (* 0.5 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, log(z), y);
	double tmp;
	if (z <= 2.4e-265) {
		tmp = t_0;
	} else if (z <= 6e-104) {
		tmp = x * fma((y / x), -z, 0.5);
	} else if (z <= 3.2e-65) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (0.5 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, log(z), y)
	tmp = 0.0
	if (z <= 2.4e-265)
		tmp = t_0;
	elseif (z <= 6e-104)
		tmp = Float64(x * fma(Float64(y / x), Float64(-z), 0.5));
	elseif (z <= 3.2e-65)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(0.5 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[z, 2.4e-265], t$95$0, If[LessEqual[z, 6e-104], N[(x * N[(N[(y / x), $MachinePrecision] * (-z) + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e-65], t$95$0, N[(y * (-z) + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.4e-265 or 6.0000000000000005e-104 < z < 3.1999999999999999e-65

   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-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 N/A

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

      11. log-lowering-log.f64 69.3

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

   5. Simplified 69.3%

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

   6. Taylor expanded in z around 0

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

      1. log-lowering-log.f64 69.3

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

   8. Simplified 69.3%

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

   if 2.4e-265 < z < 6.0000000000000005e-104

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-lowering-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-+r- N/A

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

      11. +-commutative N/A

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

      12. --lowering--.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. log-lowering-log.f64 93.9

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

   7. Simplified 93.9%

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

   8. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 60.0

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

   10. Simplified 60.0%

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

   if 3.1999999999999999e-65 < 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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-+r- N/A

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

      11. +-commutative N/A

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

      12. --lowering--.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. log-lowering-log.f64 89.5

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

   7. Simplified 89.5%

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

   8. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 84.6

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

   10. Simplified 84.6%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. mul-1-neg N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. mul-1-neg N/A

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

       6. neg-lowering-neg.f64 N/A

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

       7. *-lowering-*.f64 93.2

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

   13. Simplified 93.2%

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

Alternative 3: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log z - z\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;y + y \cdot t\_0\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, t\_0, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log z) z)))
   (if (<= y -8.5e+54)
     (+ y (* y t_0))
     (if (<= y 1.5e+58) (fma y (- z) (* 0.5 x)) (fma y t_0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = log(z) - z;
	double tmp;
	if (y <= -8.5e+54) {
		tmp = y + (y * t_0);
	} else if (y <= 1.5e+58) {
		tmp = fma(y, -z, (0.5 * x));
	} else {
		tmp = fma(y, t_0, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(z) - z)
	tmp = 0.0
	if (y <= -8.5e+54)
		tmp = Float64(y + Float64(y * t_0));
	elseif (y <= 1.5e+58)
		tmp = fma(y, Float64(-z), Float64(0.5 * x));
	else
		tmp = fma(y, t_0, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, -8.5e+54], N[(y + N[(y * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+58], N[(y * (-z) + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], N[(y * t$95$0 + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.4999999999999995e54

   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 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-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 N/A

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

      11. log-lowering-log.f64 95.3

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

   5. Simplified 95.3%

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

      1. +-lowering-+.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 95.3

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

   7. Applied egg-rr 95.3%

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

   if -8.4999999999999995e54 < y < 1.5000000000000001e58

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-+r- N/A

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

      11. +-commutative N/A

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

      12. --lowering--.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. log-lowering-log.f64 94.9

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

   7. Simplified 94.9%

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

   8. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 82.3

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

   10. Simplified 82.3%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. mul-1-neg N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. mul-1-neg N/A

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

       6. neg-lowering-neg.f64 N/A

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

       7. *-lowering-*.f64 87.3

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

   13. Simplified 87.3%

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

   if 1.5000000000000001e58 < 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-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 N/A

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

      11. log-lowering-log.f64 87.4

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

   5. Simplified 87.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+54}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- (log z) z) y)))
   (if (<= y -2.6e+65) t_0 (if (<= y 1.25e+58) (fma y (- z) (* 0.5 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (log(z) - z), y);
	double tmp;
	if (y <= -2.6e+65) {
		tmp = t_0;
	} else if (y <= 1.25e+58) {
		tmp = fma(y, -z, (0.5 * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(log(z) - z), y)
	tmp = 0.0
	if (y <= -2.6e+65)
		tmp = t_0;
	elseif (y <= 1.25e+58)
		tmp = fma(y, Float64(-z), Float64(0.5 * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -2.6e+65], t$95$0, If[LessEqual[y, 1.25e+58], N[(y * (-z) + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.60000000000000003e65 or 1.24999999999999996e58 < 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 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-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. --lowering--.f64 N/A

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

      11. log-lowering-log.f64 90.9

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

   5. Simplified 90.9%

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

   if -2.60000000000000003e65 < y < 1.24999999999999996e58

   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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. associate-+r- N/A

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

      11. +-commutative N/A

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

      12. --lowering--.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. log-lowering-log.f64 94.9

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

   7. Simplified 94.9%

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

   8. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 82.3

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

   10. Simplified 82.3%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. mul-1-neg N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. mul-1-neg N/A

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

       6. neg-lowering-neg.f64 N/A

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

       7. *-lowering-*.f64 87.3

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

   13. Simplified 87.3%

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

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(y, \log z, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, 0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.4e-6) (fma 0.5 x (fma y (log z) y)) (fma (- 1.0 z) y (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e-6) {
		tmp = fma(0.5, x, fma(y, log(z), y));
	} else {
		tmp = fma((1.0 - z), y, (0.5 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.4e-6)
		tmp = fma(0.5, x, fma(y, log(z), y));
	else
		tmp = fma(Float64(1.0 - z), y, Float64(0.5 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.4e-6], N[(0.5 * x + N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.3999999999999999e-6

   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

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 N/A

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

      13. accelerator-lowering-fma.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 99.2

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

   8. Simplified 99.2%

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

   if 2.3999999999999999e-6 < 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. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   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. *-lowering-*.f64 98.6

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

   7. Simplified 98.6%

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

Alternative 6: 61.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 3.5e+32) (* 0.5 x) (- (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.5e+32) {
		tmp = 0.5 * x;
	} else {
		tmp = -(y * z);
	}
	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 <= 3.5d+32) then
        tmp = 0.5d0 * x
    else
        tmp = -(y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.5e+32) {
		tmp = 0.5 * x;
	} else {
		tmp = -(y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.5e+32:
		tmp = 0.5 * x
	else:
		tmp = -(y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.5e+32)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(-Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.5e+32)
		tmp = 0.5 * x;
	else
		tmp = -(y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.5e+32], N[(0.5 * x), $MachinePrecision], (-N[(y * z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < 3.5000000000000001e32

   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 inf

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

      1. *-lowering-*.f64 52.0

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

   5. Simplified 52.0%

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

   if 3.5000000000000001e32 < 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. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

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

      4. neg-lowering-neg.f64 71.1

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

   5. Simplified 71.1%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+32}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+l+ N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. log-lowering-log.f64 N/A

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

   9. *-lowering-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. distribute-rgt-out N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. associate-+r- N/A

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

   11. +-commutative N/A

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

   12. --lowering--.f64 N/A

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

   13. +-lowering-+.f64 N/A

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

   14. log-lowering-log.f64 89.9

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

7. Simplified 89.9%

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

8. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 69.7

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

10. Simplified 69.7%

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

11. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

    2. distribute-rgt-neg-in N/A

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

    3. mul-1-neg N/A

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

    4. accelerator-lowering-fma.f64 N/A

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

    5. mul-1-neg N/A

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

    6. neg-lowering-neg.f64 N/A

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

    7. *-lowering-*.f64 74.5

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

13. Simplified 74.5%

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

Alternative 8: 40.2% 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 x around inf

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

   1. *-lowering-*.f64 42.0

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

5. Simplified 42.0%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Add Preprocessing

Alternative 9: 1.8% accurate, 120.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]
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-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. sub-neg N/A

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

   10. --lowering--.f64 N/A

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

   11. log-lowering-log.f64 58.6

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

5. Simplified 58.6%

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

6. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 33.0

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

8. Simplified 33.0%

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

9. Taylor expanded in z around 0

   \[\leadsto \color{blue}{y} \]
10. Step-by-step derivation

11. Simplified 2.0%

    \[\leadsto \color{blue}{y} \]
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 2024198 
(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)))))