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

Percentage Accurate: 99.9% → 99.9%

Time: 12.9s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. +-commutative N/A

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

   5. associate-+r- N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- z) y (* x 0.5))))
   (if (<= (* x 0.5) -1e-88)
     t_0
     (if (<= (* x 0.5) 2e-11) (+ y (* y (- (log z) z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-z, y, (x * 0.5));
	double tmp;
	if ((x * 0.5) <= -1e-88) {
		tmp = t_0;
	} else if ((x * 0.5) <= 2e-11) {
		tmp = y + (y * (log(z) - z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-z), y, Float64(x * 0.5))
	tmp = 0.0
	if (Float64(x * 0.5) <= -1e-88)
		tmp = t_0;
	elseif (Float64(x * 0.5) <= 2e-11)
		tmp = Float64(y + Float64(y * Float64(log(z) - z)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-88], t$95$0, If[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-11], N[(y + N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -9.99999999999999934e-89 or 1.99999999999999988e-11 < (*.f64 x #s(literal 1/2 binary64))

   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. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 88.1

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

   7. Simplified 88.1%

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

   if -9.99999999999999934e-89 < (*.f64 x #s(literal 1/2 binary64)) < 1.99999999999999988e-11

   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 92.7

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

   5. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z - z, y\right)} \]
   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 92.7

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

   7. Applied egg-rr 92.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot 0.5 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- z) y (* x 0.5))))
   (if (<= (* x 0.5) -1e-88)
     t_0
     (if (<= (* x 0.5) 2e-11) (fma y (- (log z) z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-z, y, (x * 0.5));
	double tmp;
	if ((x * 0.5) <= -1e-88) {
		tmp = t_0;
	} else if ((x * 0.5) <= 2e-11) {
		tmp = fma(y, (log(z) - z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-z), y, Float64(x * 0.5))
	tmp = 0.0
	if (Float64(x * 0.5) <= -1e-88)
		tmp = t_0;
	elseif (Float64(x * 0.5) <= 2e-11)
		tmp = fma(y, Float64(log(z) - z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-88], t$95$0, If[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-11], N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -9.99999999999999934e-89 or 1.99999999999999988e-11 < (*.f64 x #s(literal 1/2 binary64))

   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. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 88.1

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

   7. Simplified 88.1%

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

   if -9.99999999999999934e-89 < (*.f64 x #s(literal 1/2 binary64)) < 1.99999999999999988e-11

   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 92.7

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

   5. Simplified 92.7%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.27:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, \mathsf{fma}\left(0.5, x, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.27) (fma y (log z) (fma 0.5 x y)) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.27) {
		tmp = fma(y, log(z), fma(0.5, x, y));
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.27)
		tmp = fma(y, log(z), fma(0.5, x, y));
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.27], N[(y * N[Log[z], $MachinePrecision] + N[(0.5 * x + y), $MachinePrecision]), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.27000000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. accelerator-lowering-fma.f64 98.2

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

   5. Simplified 98.2%

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

   if 0.27000000000000002 < 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. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 99.1

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

   7. Simplified 99.1%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

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 6: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.6 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 8.6e-50) (fma y (log z) y) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 8.6e-50) {
		tmp = fma(y, log(z), y);
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 8.6e-50)
		tmp = fma(y, log(z), y);
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 8.6e-50], N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.59999999999999995e-50

   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 60.1

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

   5. Simplified 60.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. log-lowering-log.f64 60.1

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

   8. Simplified 60.1%

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

   if 8.59999999999999995e-50 < 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. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 94.2

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

   7. Simplified 94.2%

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

Alternative 7: 61.0% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+22}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.45e+22) (* x 0.5) (fma y (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.45e+22) {
		tmp = x * 0.5;
	} else {
		tmp = fma(y, -z, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.45e+22)
		tmp = Float64(x * 0.5);
	else
		tmp = fma(y, Float64(-z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.45e+22], N[(x * 0.5), $MachinePrecision], N[(y * (-z) + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.45e22

   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 43.3

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

   5. Simplified 43.3%

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

   if 1.45e22 < 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 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 78.0

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

   5. Simplified 78.0%

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

   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 78.0

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

   8. Simplified 78.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+22}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{+23}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.6e+23) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.6e+23) {
		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 <= 2.6d+23) 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 <= 2.6e+23) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.6e+23:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.6e+23)
		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 <= 2.6e+23)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.6e+23], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.59999999999999992e23

   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 43.3

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

   5. Simplified 43.3%

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

   if 2.59999999999999992e23 < 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 78.0

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

   5. Simplified 78.0%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{+23}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

   4. +-commutative N/A

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

   5. associate-+r- N/A

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

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

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

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

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

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

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

   9. *-lowering-*.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 71.7

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

7. Simplified 71.7%

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

Alternative 10: 40.0% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 34.2

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

5. Simplified 34.2%

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

6. Final simplification 34.2%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Alternative 11: 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 67.8

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

5. Simplified 67.8%

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

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 39.2

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

8. Simplified 39.2%

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

9. Taylor expanded in z around 0

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

11. Simplified 1.8%

    \[\leadsto \color{blue}{y} \]
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 2024199 
(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)))))