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

Percentage Accurate: 99.9% → 99.9%

Time: 12.3s

Alternatives: 11

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + 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. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+128} \lor \neg \left(y \leq 4.4 \cdot 10^{+114}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e+128) (not (<= y 4.4e+114)))
   (* y (- (+ 1.0 (log z)) z))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e+128) || !(y <= 4.4e+114)) {
		tmp = y * ((1.0 + log(z)) - z);
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e+128) || !(y <= 4.4e+114))
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e+128], N[Not[LessEqual[y, 4.4e+114]], $MachinePrecision]], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000017e128 or 4.4000000000000001e114 < 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 inf 76.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 76.9%

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

      2. +-commutative 76.9%

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

      3. associate--l+ 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in x around inf 76.9%

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

   7. Taylor expanded in x around 0 92.8%

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

   if -2.20000000000000017e128 < y < 4.4000000000000001e114

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 85.2%

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

      1. neg-mul-1 85.2%

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

   7. Simplified 85.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+128} \lor \neg \left(y \leq 4.4 \cdot 10^{+114}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;x \cdot 0.5 + x \cdot \left(\left(z + -1\right) \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.9e-22)
   (+ (* x 0.5) (* x (* (+ z -1.0) (/ y x))))
   (if (<= z 3.8e-9) (* y (+ 1.0 (log z))) (fma y (- z) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.9e-22) {
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)));
	} else if (z <= 3.8e-9) {
		tmp = y * (1.0 + log(z));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.9e-22)
		tmp = Float64(Float64(x * 0.5) + Float64(x * Float64(Float64(z + -1.0) * Float64(y / x))));
	elseif (z <= 3.8e-9)
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.9e-22], N[(N[(x * 0.5), $MachinePrecision] + N[(x * N[(N[(z + -1.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-9], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.9000000000000002e-22

   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 90.3%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 90.3%

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

      2. distribute-rgt-neg-in 90.3%

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

      3. sub-neg 90.3%

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

      4. mul-1-neg 90.3%

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

      5. *-commutative 90.3%

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

      6. +-commutative 90.3%

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

      7. associate--l+ 90.3%

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

      8. +-commutative 90.3%

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

      9. associate-/l* 90.3%

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

      10. distribute-lft-neg-in 90.3%

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

      11. associate-+l- 90.3%

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

      12. sub-neg 90.3%

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

      13. distribute-neg-in 90.3%

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

      14. metadata-eval 90.3%

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

      15. remove-double-neg 90.3%

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

   5. Simplified 90.3%

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

   6. Taylor expanded in z around inf 52.6%

      \[\leadsto x \cdot \left(-\left(\left(-1 + \color{blue}{z}\right) \cdot \frac{y}{x} + -0.5\right)\right) \]
   7. Step-by-step derivation

      1. distribute-neg-in 52.6%

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

      2. metadata-eval 52.6%

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

      3. distribute-rgt-in 52.6%

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

      4. distribute-rgt-neg-in 52.6%

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

      5. add-sqr-sqrt 28.7%

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

      6. sqrt-unprod 54.8%

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

      7. sqr-neg 54.8%

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

      8. sqrt-unprod 27.1%

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

      9. add-sqr-sqrt 58.0%

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

      10. distribute-frac-neg2 58.0%

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

      11. frac-2neg 58.0%

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

   8. Applied egg-rr 58.0%

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

   if 2.9000000000000002e-22 < z < 3.80000000000000011e-9

   1. Initial program 99.3%

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

   3. Taylor expanded in x around -inf 72.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.9%

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

      2. distribute-rgt-neg-in 72.9%

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

      3. sub-neg 72.9%

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

      4. mul-1-neg 72.9%

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

      5. *-commutative 72.9%

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

      6. +-commutative 72.9%

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

      7. associate--l+ 72.9%

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

      8. +-commutative 72.9%

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

      9. associate-/l* 72.7%

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

      10. distribute-lft-neg-in 72.7%

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

      11. associate-+l- 72.7%

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

      12. sub-neg 72.7%

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

      13. distribute-neg-in 72.7%

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

      14. metadata-eval 72.7%

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

      15. remove-double-neg 72.7%

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

   5. Simplified 72.7%

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

   6. Taylor expanded in x around 0 89.2%

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

   7. Taylor expanded in z around 0 84.4%

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

      1. neg-mul-1 84.4%

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

      2. distribute-rgt-neg-in 84.4%

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

      3. distribute-neg-in 84.4%

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

      4. metadata-eval 84.4%

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

      5. sub-neg 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in y around 0 84.4%

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

   if 3.80000000000000011e-9 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 98.9%

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

      1. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;x \cdot 0.5 + x \cdot \left(\left(z + -1\right) \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;x \cdot 0.5 + x \cdot \left(\left(z + -1\right) \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 2.95 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.8e-22)
   (+ (* x 0.5) (* x (* (+ z -1.0) (/ y x))))
   (if (<= z 2.95e-9) (* y (+ 1.0 (log z))) (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e-22) {
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)));
	} else if (z <= 2.95e-9) {
		tmp = y * (1.0 + log(z));
	} else {
		tmp = (x * 0.5) - (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 <= 1.8d-22) then
        tmp = (x * 0.5d0) + (x * ((z + (-1.0d0)) * (y / x)))
    else if (z <= 2.95d-9) then
        tmp = y * (1.0d0 + log(z))
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e-22) {
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)));
	} else if (z <= 2.95e-9) {
		tmp = y * (1.0 + Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.8e-22:
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)))
	elif z <= 2.95e-9:
		tmp = y * (1.0 + math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.8e-22)
		tmp = Float64(Float64(x * 0.5) + Float64(x * Float64(Float64(z + -1.0) * Float64(y / x))));
	elseif (z <= 2.95e-9)
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.8e-22)
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)));
	elseif (z <= 2.95e-9)
		tmp = y * (1.0 + log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.8e-22], N[(N[(x * 0.5), $MachinePrecision] + N[(x * N[(N[(z + -1.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.95e-9], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.7999999999999999e-22

   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 90.3%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 90.3%

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

      2. distribute-rgt-neg-in 90.3%

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

      3. sub-neg 90.3%

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

      4. mul-1-neg 90.3%

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

      5. *-commutative 90.3%

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

      6. +-commutative 90.3%

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

      7. associate--l+ 90.3%

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

      8. +-commutative 90.3%

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

      9. associate-/l* 90.3%

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

      10. distribute-lft-neg-in 90.3%

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

      11. associate-+l- 90.3%

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

      12. sub-neg 90.3%

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

      13. distribute-neg-in 90.3%

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

      14. metadata-eval 90.3%

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

      15. remove-double-neg 90.3%

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

   5. Simplified 90.3%

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

   6. Taylor expanded in z around inf 52.6%

      \[\leadsto x \cdot \left(-\left(\left(-1 + \color{blue}{z}\right) \cdot \frac{y}{x} + -0.5\right)\right) \]
   7. Step-by-step derivation

      1. distribute-neg-in 52.6%

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

      2. metadata-eval 52.6%

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

      3. distribute-rgt-in 52.6%

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

      4. distribute-rgt-neg-in 52.6%

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

      5. add-sqr-sqrt 28.7%

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

      6. sqrt-unprod 54.8%

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

      7. sqr-neg 54.8%

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

      8. sqrt-unprod 27.1%

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

      9. add-sqr-sqrt 58.0%

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

      10. distribute-frac-neg2 58.0%

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

      11. frac-2neg 58.0%

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

   8. Applied egg-rr 58.0%

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

   if 1.7999999999999999e-22 < z < 2.9499999999999999e-9

   1. Initial program 99.3%

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

   3. Taylor expanded in x around -inf 72.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.9%

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

      2. distribute-rgt-neg-in 72.9%

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

      3. sub-neg 72.9%

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

      4. mul-1-neg 72.9%

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

      5. *-commutative 72.9%

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

      6. +-commutative 72.9%

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

      7. associate--l+ 72.9%

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

      8. +-commutative 72.9%

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

      9. associate-/l* 72.7%

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

      10. distribute-lft-neg-in 72.7%

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

      11. associate-+l- 72.7%

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

      12. sub-neg 72.7%

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

      13. distribute-neg-in 72.7%

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

      14. metadata-eval 72.7%

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

      15. remove-double-neg 72.7%

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

   5. Simplified 72.7%

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

   6. Taylor expanded in x around 0 89.2%

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

   7. Taylor expanded in z around 0 84.4%

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

      1. neg-mul-1 84.4%

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

      2. distribute-rgt-neg-in 84.4%

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

      3. distribute-neg-in 84.4%

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

      4. metadata-eval 84.4%

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

      5. sub-neg 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in y around 0 84.4%

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

   if 2.9499999999999999e-9 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 98.9%

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

      1. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + 0.5 \cdot x} \]
   9. Step-by-step derivation

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. mul-1-neg 98.9%

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

      4. unsub-neg 98.9%

         \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]

      5. *-commutative 98.9%

         \[\leadsto \color{blue}{0.5 \cdot x} - y \cdot z \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{0.5 \cdot x - y \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;x \cdot 0.5 + x \cdot \left(\left(z + -1\right) \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 2.95 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.027:\\ \;\;\;\;y \cdot \left(1 + \log z\right) + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.027)
   (+ (* y (+ 1.0 (log z))) (* x 0.5))
   (- (* x 0.5) (* y (+ z -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.027) {
		tmp = (y * (1.0 + log(z))) + (x * 0.5);
	} else {
		tmp = (x * 0.5) - (y * (z + -1.0));
	}
	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 <= 0.027d0) then
        tmp = (y * (1.0d0 + log(z))) + (x * 0.5d0)
    else
        tmp = (x * 0.5d0) - (y * (z + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.027) {
		tmp = (y * (1.0 + Math.log(z))) + (x * 0.5);
	} else {
		tmp = (x * 0.5) - (y * (z + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.027:
		tmp = (y * (1.0 + math.log(z))) + (x * 0.5)
	else:
		tmp = (x * 0.5) - (y * (z + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.027)
		tmp = Float64(Float64(y * Float64(1.0 + log(z))) + Float64(x * 0.5));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * Float64(z + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.027)
		tmp = (y * (1.0 + log(z))) + (x * 0.5);
	else
		tmp = (x * 0.5) - (y * (z + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.027], N[(N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0269999999999999997

   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 98.7%

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

   if 0.0269999999999999997 < 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 x around -inf 85.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 85.9%

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

      2. distribute-rgt-neg-in 85.9%

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

      3. sub-neg 85.9%

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

      4. mul-1-neg 85.9%

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

      5. *-commutative 85.9%

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

      6. +-commutative 85.9%

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

      7. associate--l+ 85.9%

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

      8. +-commutative 85.9%

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

      9. associate-/l* 85.8%

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

      10. distribute-lft-neg-in 85.8%

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

      11. associate-+l- 85.8%

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

      12. sub-neg 85.8%

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

      13. distribute-neg-in 85.8%

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

      14. metadata-eval 85.8%

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

      15. remove-double-neg 85.8%

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

   5. Simplified 85.8%

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

   6. Taylor expanded in z around inf 85.7%

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

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z - 1\right)\right) + 0.5 \cdot x} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

   9. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot x - y \cdot \left(-1 + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.027:\\ \;\;\;\;y \cdot \left(1 + \log z\right) + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y * ((1.0 - z) + log(z))) + (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 = (y * ((1.0d0 - z) + log(z))) + (x * 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 76.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.027:\\ \;\;\;\;x \cdot 0.5 + x \cdot \left(\left(z + -1\right) \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.027)
   (+ (* x 0.5) (* x (* (+ z -1.0) (/ y x))))
   (- (* x 0.5) (* y (+ z -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.027) {
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)));
	} else {
		tmp = (x * 0.5) - (y * (z + -1.0));
	}
	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 <= 0.027d0) then
        tmp = (x * 0.5d0) + (x * ((z + (-1.0d0)) * (y / x)))
    else
        tmp = (x * 0.5d0) - (y * (z + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.027) {
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)));
	} else {
		tmp = (x * 0.5) - (y * (z + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.027:
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)))
	else:
		tmp = (x * 0.5) - (y * (z + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.027)
		tmp = Float64(Float64(x * 0.5) + Float64(x * Float64(Float64(z + -1.0) * Float64(y / x))));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * Float64(z + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.027)
		tmp = (x * 0.5) + (x * ((z + -1.0) * (y / x)));
	else
		tmp = (x * 0.5) - (y * (z + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.027], N[(N[(x * 0.5), $MachinePrecision] + N[(x * N[(N[(z + -1.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0269999999999999997

   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 89.7%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 89.7%

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

      2. distribute-rgt-neg-in 89.7%

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

      3. sub-neg 89.7%

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

      4. mul-1-neg 89.7%

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

      5. *-commutative 89.7%

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

      6. +-commutative 89.7%

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

      7. associate--l+ 89.7%

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

      8. +-commutative 89.7%

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

      9. associate-/l* 89.7%

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

      10. distribute-lft-neg-in 89.7%

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

      11. associate-+l- 89.7%

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

      12. sub-neg 89.7%

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

      13. distribute-neg-in 89.7%

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

      14. metadata-eval 89.7%

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

      15. remove-double-neg 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in z around inf 50.5%

      \[\leadsto x \cdot \left(-\left(\left(-1 + \color{blue}{z}\right) \cdot \frac{y}{x} + -0.5\right)\right) \]
   7. Step-by-step derivation

      1. distribute-neg-in 50.5%

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

      2. metadata-eval 50.5%

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

      3. distribute-rgt-in 50.5%

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

      4. distribute-rgt-neg-in 50.5%

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

      5. add-sqr-sqrt 27.8%

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

      6. sqrt-unprod 52.7%

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

      7. sqr-neg 52.7%

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

      8. sqrt-unprod 25.9%

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

      9. add-sqr-sqrt 56.2%

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

      10. distribute-frac-neg2 56.2%

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

      11. frac-2neg 56.2%

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

   8. Applied egg-rr 56.2%

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

   if 0.0269999999999999997 < 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 x around -inf 85.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 85.9%

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

      2. distribute-rgt-neg-in 85.9%

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

      3. sub-neg 85.9%

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

      4. mul-1-neg 85.9%

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

      5. *-commutative 85.9%

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

      6. +-commutative 85.9%

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

      7. associate--l+ 85.9%

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

      8. +-commutative 85.9%

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

      9. associate-/l* 85.8%

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

      10. distribute-lft-neg-in 85.8%

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

      11. associate-+l- 85.8%

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

      12. sub-neg 85.8%

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

      13. distribute-neg-in 85.8%

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

      14. metadata-eval 85.8%

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

      15. remove-double-neg 85.8%

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

   5. Simplified 85.8%

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

   6. Taylor expanded in z around inf 85.7%

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

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z - 1\right)\right) + 0.5 \cdot x} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

   9. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot x - y \cdot \left(-1 + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.027:\\ \;\;\;\;x \cdot 0.5 + x \cdot \left(\left(z + -1\right) \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.027:\\ \;\;\;\;x \cdot \left(\left(z + -1\right) \cdot \frac{y}{x} - -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.027)
   (* x (- (* (+ z -1.0) (/ y x)) -0.5))
   (- (* x 0.5) (* y (+ z -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.027) {
		tmp = x * (((z + -1.0) * (y / x)) - -0.5);
	} else {
		tmp = (x * 0.5) - (y * (z + -1.0));
	}
	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 <= 0.027d0) then
        tmp = x * (((z + (-1.0d0)) * (y / x)) - (-0.5d0))
    else
        tmp = (x * 0.5d0) - (y * (z + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.027) {
		tmp = x * (((z + -1.0) * (y / x)) - -0.5);
	} else {
		tmp = (x * 0.5) - (y * (z + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.027:
		tmp = x * (((z + -1.0) * (y / x)) - -0.5)
	else:
		tmp = (x * 0.5) - (y * (z + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.027)
		tmp = Float64(x * Float64(Float64(Float64(z + -1.0) * Float64(y / x)) - -0.5));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * Float64(z + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.027)
		tmp = x * (((z + -1.0) * (y / x)) - -0.5);
	else
		tmp = (x * 0.5) - (y * (z + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.027], N[(x * N[(N[(N[(z + -1.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0269999999999999997

   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 89.7%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 89.7%

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

      2. distribute-rgt-neg-in 89.7%

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

      3. sub-neg 89.7%

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

      4. mul-1-neg 89.7%

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

      5. *-commutative 89.7%

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

      6. +-commutative 89.7%

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

      7. associate--l+ 89.7%

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

      8. +-commutative 89.7%

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

      9. associate-/l* 89.7%

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

      10. distribute-lft-neg-in 89.7%

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

      11. associate-+l- 89.7%

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

      12. sub-neg 89.7%

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

      13. distribute-neg-in 89.7%

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

      14. metadata-eval 89.7%

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

      15. remove-double-neg 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in z around inf 50.5%

      \[\leadsto x \cdot \left(-\left(\left(-1 + \color{blue}{z}\right) \cdot \frac{y}{x} + -0.5\right)\right) \]
   7. Step-by-step derivation

      1. distribute-neg-in 50.5%

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

      2. distribute-rgt-neg-in 50.5%

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

      3. add-sqr-sqrt 27.8%

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

      4. sqrt-unprod 52.7%

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

      5. sqr-neg 52.7%

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

      6. sqrt-unprod 25.9%

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

      7. add-sqr-sqrt 56.2%

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

      8. distribute-frac-neg2 56.2%

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

      9. frac-2neg 56.2%

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

      10. sub-neg 56.2%

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

   8. Applied egg-rr 56.2%

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

   if 0.0269999999999999997 < 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 x around -inf 85.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x} - 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 85.9%

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

      2. distribute-rgt-neg-in 85.9%

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

      3. sub-neg 85.9%

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

      4. mul-1-neg 85.9%

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

      5. *-commutative 85.9%

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

      6. +-commutative 85.9%

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

      7. associate--l+ 85.9%

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

      8. +-commutative 85.9%

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

      9. associate-/l* 85.8%

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

      10. distribute-lft-neg-in 85.8%

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

      11. associate-+l- 85.8%

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

      12. sub-neg 85.8%

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

      13. distribute-neg-in 85.8%

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

      14. metadata-eval 85.8%

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

      15. remove-double-neg 85.8%

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

   5. Simplified 85.8%

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

   6. Taylor expanded in z around inf 85.7%

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

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z - 1\right)\right) + 0.5 \cdot x} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. sub-neg 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

      6. +-commutative 99.6%

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

   9. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot x - y \cdot \left(-1 + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.027:\\ \;\;\;\;x \cdot \left(\left(z + -1\right) \cdot \frac{y}{x} - -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{+68}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 8.5e+68) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 8.5e+68) {
		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 <= 8.5d+68) 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 <= 8.5e+68) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 8.5e+68:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 8.5e+68)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 8.5e+68)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 8.5e+68], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.49999999999999966e68

   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 53.4%

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

   if 8.49999999999999966e68 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around inf 100.0%

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot y + 0.5 \cdot \frac{x}{z}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

      3. unsub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 78.7%

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

       1. neg-mul-1 78.7%

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

   13. Simplified 78.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{+68}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 15.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * 0.5) - (y * 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 * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in z around inf 75.9%

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

   1. neg-mul-1 75.9%

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

7. Simplified 75.9%

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

8. Taylor expanded in y around 0 75.9%

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + 0.5 \cdot x} \]
9. Step-by-step derivation

   1. +-commutative 75.9%

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

   2. *-commutative 75.9%

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

   3. mul-1-neg 75.9%

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

   4. unsub-neg 75.9%

      \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]

   5. *-commutative 75.9%

      \[\leadsto \color{blue}{0.5 \cdot x} - y \cdot z \]

10. Simplified 75.9%

    \[\leadsto \color{blue}{0.5 \cdot x - y \cdot z} \]

11. Final simplification 75.9%

    \[\leadsto x \cdot 0.5 - y \cdot z \]
12. Add Preprocessing

Alternative 11: 40.0% accurate, 37.0× 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 x around inf 40.8%

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

4. Final simplification 40.8%

   \[\leadsto x \cdot 0.5 \]
5. 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 2024114 
(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)))))