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

Percentage Accurate: 99.9% → 99.9%

Time: 11.4s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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. Final simplification 99.9%

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

Alternative 2: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ \mathbf{if}\;z \leq 5.4 \cdot 10^{-295}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-216}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-96}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))))
   (if (<= z 5.4e-295)
     (* x 0.5)
     (if (<= z 2.9e-280)
       t_0
       (if (<= z 3.5e-216)
         (- (* x 0.5) (* y z))
         (if (<= z 1.55e-96) t_0 (fma y (- z) (* x 0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double tmp;
	if (z <= 5.4e-295) {
		tmp = x * 0.5;
	} else if (z <= 2.9e-280) {
		tmp = t_0;
	} else if (z <= 3.5e-216) {
		tmp = (x * 0.5) - (y * z);
	} else if (z <= 1.55e-96) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	tmp = 0.0
	if (z <= 5.4e-295)
		tmp = Float64(x * 0.5);
	elseif (z <= 2.9e-280)
		tmp = t_0;
	elseif (z <= 3.5e-216)
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	elseif (z <= 1.55e-96)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.4e-295], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 2.9e-280], t$95$0, If[LessEqual[z, 3.5e-216], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e-96], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 5.4000000000000002e-295

   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 100.0%

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

   if 5.4000000000000002e-295 < z < 2.9e-280 or 3.49999999999999982e-216 < z < 1.55e-96

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 53.4%

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

      4. associate-*r* 53.4%

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

      5. fma-define 53.4%

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

   4. Applied egg-rr 53.4%

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

   5. Taylor expanded in y around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. +-commutative 0.0%

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

      4. associate-+r- 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 70.3%

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

      8. *-commutative 70.3%

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

      9. neg-mul-1 70.3%

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

      10. *-commutative 70.3%

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

      11. sub-neg 70.3%

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

      12. associate-+l+ 70.3%

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

      13. +-commutative 70.3%

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

      14. distribute-lft-in 70.3%

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

      15. neg-mul-1 70.3%

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

      16. remove-double-neg 70.3%

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

      17. neg-mul-1 70.3%

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

      18. +-commutative 70.3%

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

      19. distribute-neg-in 70.3%

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

      20. metadata-eval 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in z around 0 70.3%

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

   if 2.9e-280 < z < 3.49999999999999982e-216

   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 inf 61.4%

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

      1. mul-1-neg 61.4%

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

      2. *-commutative 61.4%

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

      3. distribute-rgt-neg-in 61.4%

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

   5. Simplified 61.4%

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

      1. distribute-rgt-neg-out 61.4%

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

      2. unsub-neg 61.4%

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

      3. *-commutative 61.4%

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

   7. Applied egg-rr 61.4%

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

   if 1.55e-96 < z

   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 87.5%

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

      1. mul-1-neg 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-295}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-280}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-216}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-96}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.8 \cdot 10^{-296}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-280} \lor \neg \left(z \leq 2 \cdot 10^{-216}\right) \land z \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 8.8e-296)
   (* x 0.5)
   (if (or (<= z 7.2e-280) (and (not (<= z 2e-216)) (<= z 1.6e-96)))
     (* y (+ 1.0 (log z)))
     (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 8.8e-296) {
		tmp = x * 0.5;
	} else if ((z <= 7.2e-280) || (!(z <= 2e-216) && (z <= 1.6e-96))) {
		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 <= 8.8d-296) then
        tmp = x * 0.5d0
    else if ((z <= 7.2d-280) .or. (.not. (z <= 2d-216)) .and. (z <= 1.6d-96)) 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 <= 8.8e-296) {
		tmp = x * 0.5;
	} else if ((z <= 7.2e-280) || (!(z <= 2e-216) && (z <= 1.6e-96))) {
		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 <= 8.8e-296:
		tmp = x * 0.5
	elif (z <= 7.2e-280) or (not (z <= 2e-216) and (z <= 1.6e-96)):
		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 <= 8.8e-296)
		tmp = Float64(x * 0.5);
	elseif ((z <= 7.2e-280) || (!(z <= 2e-216) && (z <= 1.6e-96)))
		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 <= 8.8e-296)
		tmp = x * 0.5;
	elseif ((z <= 7.2e-280) || (~((z <= 2e-216)) && (z <= 1.6e-96)))
		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, 8.8e-296], N[(x * 0.5), $MachinePrecision], If[Or[LessEqual[z, 7.2e-280], And[N[Not[LessEqual[z, 2e-216]], $MachinePrecision], LessEqual[z, 1.6e-96]]], 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 < 8.80000000000000048e-296

   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 100.0%

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

   if 8.80000000000000048e-296 < z < 7.19999999999999989e-280 or 2.0000000000000001e-216 < z < 1.60000000000000006e-96

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 53.4%

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

      4. associate-*r* 53.4%

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

      5. fma-define 53.4%

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

   4. Applied egg-rr 53.4%

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

   5. Taylor expanded in y around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. +-commutative 0.0%

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

      4. associate-+r- 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 70.3%

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

      8. *-commutative 70.3%

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

      9. neg-mul-1 70.3%

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

      10. *-commutative 70.3%

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

      11. sub-neg 70.3%

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

      12. associate-+l+ 70.3%

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

      13. +-commutative 70.3%

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

      14. distribute-lft-in 70.3%

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

      15. neg-mul-1 70.3%

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

      16. remove-double-neg 70.3%

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

      17. neg-mul-1 70.3%

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

      18. +-commutative 70.3%

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

      19. distribute-neg-in 70.3%

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

      20. metadata-eval 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in z around 0 70.3%

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

   if 7.19999999999999989e-280 < z < 2.0000000000000001e-216 or 1.60000000000000006e-96 < 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 84.2%

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

      1. mul-1-neg 84.2%

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

      2. *-commutative 84.2%

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

      3. distribute-rgt-neg-in 84.2%

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

   5. Simplified 84.2%

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

      1. distribute-rgt-neg-out 84.2%

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

      2. unsub-neg 84.2%

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

      3. *-commutative 84.2%

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

   7. Applied egg-rr 84.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.8 \cdot 10^{-296}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-280} \lor \neg \left(z \leq 2 \cdot 10^{-216}\right) \land z \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+105} \lor \neg \left(y \leq 2.4 \cdot 10^{+55}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.8e+105) (not (<= y 2.4e+55)))
   (* 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 <= -6.8e+105) || !(y <= 2.4e+55)) {
		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 <= -6.8e+105) || !(y <= 2.4e+55))
		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, -6.8e+105], N[Not[LessEqual[y, 2.4e+55]], $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 < -6.7999999999999999e105 or 2.3999999999999999e55 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-cube-cbrt 98.5%

         \[\leadsto \left(\left(1 - z\right) + \log z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + x \cdot 0.5 \]

      4. associate-*r* 98.5%

         \[\leadsto \color{blue}{\left(\left(\left(1 - z\right) + \log z\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}} + x \cdot 0.5 \]

      5. fma-define 98.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(1 - z\right) + \log z\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right), \sqrt[3]{y}, x \cdot 0.5\right)} \]

      6. pow2 98.5%

         \[\leadsto \mathsf{fma}\left(\left(\left(1 - z\right) + \log z\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}, \sqrt[3]{y}, x \cdot 0.5\right) \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(1 - z\right) + \log z\right) \cdot {\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y}, x \cdot 0.5\right)} \]

   5. Taylor expanded in y around inf 93.6%

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

   if -6.7999999999999999e105 < y < 2.3999999999999999e55

   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 86.4%

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

      1. mul-1-neg 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+105} \lor \neg \left(y \leq 2.4 \cdot 10^{+55}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.28) (+ (* x 0.5) (* y (+ 1.0 (log z)))) (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (x * 0.5) + (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 <= 0.28)
		tmp = Float64(Float64(x * 0.5) + 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, 0.28], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   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 98.6%

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

   if 0.28000000000000003 < 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 99.5%

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

      1. mul-1-neg 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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]

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%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $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. +-commutative 99.9%

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

   2. associate-+r- 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 60.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+33} \lor \neg \left(z \leq 6 \cdot 10^{+73}\right) \land z \leq 1.45 \cdot 10^{+96}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 1.7e+33) (and (not (<= z 6e+73)) (<= z 1.45e+96)))
   (* x 0.5)
   (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1.7e+33) || (!(z <= 6e+73) && (z <= 1.45e+96))) {
		tmp = x * 0.5;
	} 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 <= 1.7d+33) .or. (.not. (z <= 6d+73)) .and. (z <= 1.45d+96)) then
        tmp = x * 0.5d0
    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 <= 1.7e+33) || (!(z <= 6e+73) && (z <= 1.45e+96))) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 1.7e+33) or (not (z <= 6e+73) and (z <= 1.45e+96)):
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 1.7e+33) || (!(z <= 6e+73) && (z <= 1.45e+96)))
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 1.7e+33) || (~((z <= 6e+73)) && (z <= 1.45e+96)))
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 1.7e+33], And[N[Not[LessEqual[z, 6e+73]], $MachinePrecision], LessEqual[z, 1.45e+96]]], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.7e33 or 6.00000000000000021e73 < z < 1.44999999999999989e96

   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 49.9%

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

   if 1.7e33 < z < 6.00000000000000021e73 or 1.44999999999999989e96 < 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 100.0%

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

      2. *-commutative 100.0%

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

      3. add-sqr-sqrt 50.2%

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

      4. associate-*r* 50.2%

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

      5. fma-define 50.2%

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

   4. Applied egg-rr 50.2%

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

   5. Taylor expanded in y around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. +-commutative 0.0%

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

      4. associate-+r- 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 75.9%

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

      8. *-commutative 75.9%

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

      9. neg-mul-1 75.9%

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

      10. *-commutative 75.9%

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

      11. sub-neg 75.9%

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

      12. associate-+l+ 75.9%

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

      13. +-commutative 75.9%

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

      14. distribute-lft-in 75.9%

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

      15. neg-mul-1 75.9%

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

      16. remove-double-neg 75.9%

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

      17. neg-mul-1 75.9%

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

      18. +-commutative 75.9%

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

      19. distribute-neg-in 75.9%

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

      20. metadata-eval 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in z around inf 75.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+33} \lor \neg \left(z \leq 6 \cdot 10^{+73}\right) \land z \leq 1.45 \cdot 10^{+96}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% 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. Add Preprocessing

3. Taylor expanded in z around inf 73.9%

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

   1. mul-1-neg 73.9%

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

   2. *-commutative 73.9%

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

   3. distribute-rgt-neg-in 73.9%

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

5. Simplified 73.9%

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

   1. distribute-rgt-neg-out 73.9%

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

   2. unsub-neg 73.9%

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

   3. *-commutative 73.9%

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

7. Applied egg-rr 73.9%

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

8. Final simplification 73.9%

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

Alternative 10: 40.7% 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 39.4%

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

4. Final simplification 39.4%

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

Developer target: 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 2024035 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))