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

Percentage Accurate: 99.9% → 99.6%

Time: 9.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift--.f64 N/A

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

   9. associate-+r- N/A

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

   10. lower--.f64 N/A

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

   11. lower-+.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate--l+ N/A

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

   7. +-commutative N/A

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

   8. lift-log.f64 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. distribute-rgt-in N/A

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

   14. sub-neg N/A

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

   15. lift-neg.f64 N/A

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

   16. +-commutative N/A

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

   17. distribute-rgt1-in N/A

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

   18. lift-*.f64 N/A

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

   19. +-commutative N/A

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

   20. associate-+l+ N/A

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

   21. lift-fma.f64 N/A

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

6. Applied rewrites 99.9%

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

Alternative 2: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) + \log z\\ t_1 := \mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -220:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -108:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- 1.0 z) (log z))) (t_1 (fma (- z) y (* 0.5 x))))
   (if (<= t_0 -220.0) t_1 (if (<= t_0 -108.0) (fma (log z) y y) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 - z) + log(z);
	double t_1 = fma(-z, y, (0.5 * x));
	double tmp;
	if (t_0 <= -220.0) {
		tmp = t_1;
	} else if (t_0 <= -108.0) {
		tmp = fma(log(z), y, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - z) + log(z))
	t_1 = fma(Float64(-z), y, Float64(0.5 * x))
	tmp = 0.0
	if (t_0 <= -220.0)
		tmp = t_1;
	elseif (t_0 <= -108.0)
		tmp = fma(log(z), y, y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-z) * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -220.0], t$95$1, If[LessEqual[t$95$0, -108.0], N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -220 or -108 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.9

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

   5. Applied rewrites 81.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if -220 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -108

   1. Initial program 99.6%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   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. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.3%

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

Alternative 3: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \mathbf{if}\;0.5 \cdot x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;0.5 \cdot x \leq 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- z) y (* 0.5 x))))
   (if (<= (* 0.5 x) -4e-49)
     t_0
     (if (<= (* 0.5 x) 1e-64) (fma (- (log z) z) y y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-z, y, (0.5 * x));
	double tmp;
	if ((0.5 * x) <= -4e-49) {
		tmp = t_0;
	} else if ((0.5 * x) <= 1e-64) {
		tmp = fma((log(z) - z), y, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-z), y, Float64(0.5 * x))
	tmp = 0.0
	if (Float64(0.5 * x) <= -4e-49)
		tmp = t_0;
	elseif (Float64(0.5 * x) <= 1e-64)
		tmp = fma(Float64(log(z) - z), y, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(0.5 * x), $MachinePrecision], -4e-49], t$95$0, If[LessEqual[N[(0.5 * x), $MachinePrecision], 1e-64], N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -3.99999999999999975e-49 or 9.99999999999999965e-65 < (*.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. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 87.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

   if -3.99999999999999975e-49 < (*.f64 x #s(literal 1/2 binary64)) < 9.99999999999999965e-65

   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-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \mathbf{elif}\;0.5 \cdot x \leq 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 0.27000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

   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. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.5

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

   5. Applied rewrites 97.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.5

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

   7. Applied rewrites 97.5%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift--.f64 N/A

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

   9. associate-+r- N/A

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

   10. lower--.f64 N/A

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

   11. lower-+.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   9. lower-fma.f64 N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 7: 60.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 28.5:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 28.5) (* 0.5 x) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 28.5) {
		tmp = 0.5 * x;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 28.5d0) then
        tmp = 0.5d0 * x
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 28.5) {
		tmp = 0.5 * x;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 28.5:
		tmp = 0.5 * x
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 28.5)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 28.5)
		tmp = 0.5 * x;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 28.5], N[(0.5 * x), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 28.5

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift--.f64 N/A

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

      9. associate-+r- N/A

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

      10. lower--.f64 N/A

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

      11. lower-+.f64 99.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 93.3

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 56.5%

       \[\leadsto 0.5 \cdot x \]

   if 28.5 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 72.5

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

   5. Applied rewrites 72.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 28.5:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 77.2

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

5. Applied rewrites 77.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 77.2

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 77.2

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

7. Applied rewrites 77.2%

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

Alternative 9: 40.1% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift--.f64 N/A

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

   9. associate-+r- N/A

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

   10. lower--.f64 N/A

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

   11. lower-+.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 85.8

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

7. Applied rewrites 85.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 42.8%

    \[\leadsto 0.5 \cdot x \]
11. 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 2024332 
(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)))))