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

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. 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. fma-def 99.8%

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

   4. sub-neg 99.8%

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

   5. associate-+l+ 99.8%

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

   6. +-commutative 99.8%

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

   7. sub-neg 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\log z + 1\right)\\ t_1 := x \cdot 0.5 - z \cdot y\\ \mathbf{if}\;z \leq 3.2 \cdot 10^{-282}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-222}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-164}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (log z) 1.0))) (t_1 (- (* x 0.5) (* z y))))
   (if (<= z 3.2e-282)
     t_0
     (if (<= z 3.6e-238)
       t_1
       (if (<= z 3.5e-222)
         t_0
         (if (<= z 3.6e-191)
           t_1
           (if (<= z 9e-164)
             (+ y (* (log z) y))
             (if (<= z 1.15e-53)
               t_1
               (if (<= z 2.1e-9) t_0 (fma (- z) y (* x 0.5)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (log(z) + 1.0);
	double t_1 = (x * 0.5) - (z * y);
	double tmp;
	if (z <= 3.2e-282) {
		tmp = t_0;
	} else if (z <= 3.6e-238) {
		tmp = t_1;
	} else if (z <= 3.5e-222) {
		tmp = t_0;
	} else if (z <= 3.6e-191) {
		tmp = t_1;
	} else if (z <= 9e-164) {
		tmp = y + (log(z) * y);
	} else if (z <= 1.15e-53) {
		tmp = t_1;
	} else if (z <= 2.1e-9) {
		tmp = t_0;
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(log(z) + 1.0))
	t_1 = Float64(Float64(x * 0.5) - Float64(z * y))
	tmp = 0.0
	if (z <= 3.2e-282)
		tmp = t_0;
	elseif (z <= 3.6e-238)
		tmp = t_1;
	elseif (z <= 3.5e-222)
		tmp = t_0;
	elseif (z <= 3.6e-191)
		tmp = t_1;
	elseif (z <= 9e-164)
		tmp = Float64(y + Float64(log(z) * y));
	elseif (z <= 1.15e-53)
		tmp = t_1;
	elseif (z <= 2.1e-9)
		tmp = t_0;
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[Log[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.2e-282], t$95$0, If[LessEqual[z, 3.6e-238], t$95$1, If[LessEqual[z, 3.5e-222], t$95$0, If[LessEqual[z, 3.6e-191], t$95$1, If[LessEqual[z, 9e-164], N[(y + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e-53], t$95$1, If[LessEqual[z, 2.1e-9], t$95$0, N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 3.19999999999999983e-282 or 3.6000000000000001e-238 < z < 3.50000000000000024e-222 or 1.1500000000000001e-53 < z < 2.10000000000000019e-9

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-+l+ 99.5%

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

      3. distribute-lft-in 99.4%

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

      4. *-rgt-identity 99.4%

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

      5. associate-+r+ 99.4%

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

      6. fma-def 99.4%

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

      7. +-commutative 99.4%

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

      8. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around 0 98.1%

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

   5. Taylor expanded in y around inf 84.0%

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

   if 3.19999999999999983e-282 < z < 3.6000000000000001e-238 or 3.50000000000000024e-222 < z < 3.60000000000000019e-191 or 8.9999999999999995e-164 < z < 1.1500000000000001e-53

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 69.2%

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

      1. mul-1-neg 69.2%

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

      2. distribute-rgt-neg-out 69.2%

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

   4. Simplified 69.2%

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

      1. fma-def 69.2%

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

      2. distribute-rgt-neg-out 69.2%

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

      3. add-sqr-sqrt 69.2%

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

      4. sqrt-unprod 69.2%

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

      5. sqr-neg 69.2%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 68.6%

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

      8. fma-neg 68.6%

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

      9. *-commutative 68.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 69.2%

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

      12. sqr-neg 69.2%

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

      13. sqrt-unprod 69.2%

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

      14. add-sqr-sqrt 69.2%

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

   6. Applied egg-rr 69.2%

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

   if 3.60000000000000019e-191 < z < 8.9999999999999995e-164

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-+l+ 99.5%

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

      3. distribute-lft-in 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.7%

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

   5. Taylor expanded in x around 0 72.9%

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

   if 2.10000000000000019e-9 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 98.2%

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

      1. neg-mul-1 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{-282}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-238}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-222}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-191}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-164}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-53}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-283} \lor \neg \left(z \leq 5.6 \cdot 10^{-238}\right) \land \left(z \leq 4.6 \cdot 10^{-222} \lor \neg \left(z \leq 4.4 \cdot 10^{-191}\right) \land \left(z \leq 3 \cdot 10^{-163} \lor \neg \left(z \leq 1.05 \cdot 10^{-52}\right) \land z \leq 2.4 \cdot 10^{-9}\right)\right):\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 5.2e-283)
         (and (not (<= z 5.6e-238))
              (or (<= z 4.6e-222)
                  (and (not (<= z 4.4e-191))
                       (or (<= z 3e-163)
                           (and (not (<= z 1.05e-52)) (<= z 2.4e-9)))))))
   (* y (+ (log z) 1.0))
   (- (* x 0.5) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 5.2e-283) || (!(z <= 5.6e-238) && ((z <= 4.6e-222) || (!(z <= 4.4e-191) && ((z <= 3e-163) || (!(z <= 1.05e-52) && (z <= 2.4e-9))))))) {
		tmp = y * (log(z) + 1.0);
	} else {
		tmp = (x * 0.5) - (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 <= 5.2d-283) .or. (.not. (z <= 5.6d-238)) .and. (z <= 4.6d-222) .or. (.not. (z <= 4.4d-191)) .and. (z <= 3d-163) .or. (.not. (z <= 1.05d-52)) .and. (z <= 2.4d-9)) then
        tmp = y * (log(z) + 1.0d0)
    else
        tmp = (x * 0.5d0) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 5.2e-283) || (!(z <= 5.6e-238) && ((z <= 4.6e-222) || (!(z <= 4.4e-191) && ((z <= 3e-163) || (!(z <= 1.05e-52) && (z <= 2.4e-9))))))) {
		tmp = y * (Math.log(z) + 1.0);
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 5.2e-283) or (not (z <= 5.6e-238) and ((z <= 4.6e-222) or (not (z <= 4.4e-191) and ((z <= 3e-163) or (not (z <= 1.05e-52) and (z <= 2.4e-9)))))):
		tmp = y * (math.log(z) + 1.0)
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 5.2e-283) || (!(z <= 5.6e-238) && ((z <= 4.6e-222) || (!(z <= 4.4e-191) && ((z <= 3e-163) || (!(z <= 1.05e-52) && (z <= 2.4e-9)))))))
		tmp = Float64(y * Float64(log(z) + 1.0));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 5.2e-283) || (~((z <= 5.6e-238)) && ((z <= 4.6e-222) || (~((z <= 4.4e-191)) && ((z <= 3e-163) || (~((z <= 1.05e-52)) && (z <= 2.4e-9)))))))
		tmp = y * (log(z) + 1.0);
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 5.2e-283], And[N[Not[LessEqual[z, 5.6e-238]], $MachinePrecision], Or[LessEqual[z, 4.6e-222], And[N[Not[LessEqual[z, 4.4e-191]], $MachinePrecision], Or[LessEqual[z, 3e-163], And[N[Not[LessEqual[z, 1.05e-52]], $MachinePrecision], LessEqual[z, 2.4e-9]]]]]]], N[(y * N[(N[Log[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.2000000000000002e-283 or 5.60000000000000008e-238 < z < 4.6000000000000003e-222 or 4.39999999999999996e-191 < z < 3.0000000000000002e-163 or 1.0499999999999999e-52 < z < 2.4e-9

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-+l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. associate-+r+ 99.5%

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

      6. fma-def 99.5%

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

      7. +-commutative 99.5%

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

      8. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in z around 0 98.7%

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

   5. Taylor expanded in y around inf 79.9%

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

   if 5.2000000000000002e-283 < z < 5.60000000000000008e-238 or 4.6000000000000003e-222 < z < 4.39999999999999996e-191 or 3.0000000000000002e-163 < z < 1.0499999999999999e-52 or 2.4e-9 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 88.6%

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

      1. mul-1-neg 88.6%

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

      2. distribute-rgt-neg-out 88.6%

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

   4. Simplified 88.6%

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

      1. fma-def 88.6%

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

      2. distribute-rgt-neg-out 88.6%

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

      3. add-sqr-sqrt 88.3%

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

      4. sqrt-unprod 67.3%

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

      5. sqr-neg 67.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 39.6%

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

      8. fma-neg 39.6%

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

      9. *-commutative 39.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 67.3%

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

      12. sqr-neg 67.3%

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

      13. sqrt-unprod 88.3%

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

      14. add-sqr-sqrt 88.6%

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

   6. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{x \cdot 0.5 - z \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-283} \lor \neg \left(z \leq 5.6 \cdot 10^{-238}\right) \land \left(z \leq 4.6 \cdot 10^{-222} \lor \neg \left(z \leq 4.4 \cdot 10^{-191}\right) \land \left(z \leq 3 \cdot 10^{-163} \lor \neg \left(z \leq 1.05 \cdot 10^{-52}\right) \land z \leq 2.4 \cdot 10^{-9}\right)\right):\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 4: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\log z + 1\right)\\ t_1 := x \cdot 0.5 - z \cdot y\\ \mathbf{if}\;z \leq 6.5 \cdot 10^{-282}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-220}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-163}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-53} \lor \neg \left(z \leq 3.4 \cdot 10^{-9}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (log z) 1.0))) (t_1 (- (* x 0.5) (* z y))))
   (if (<= z 6.5e-282)
     t_0
     (if (<= z 6.6e-238)
       t_1
       (if (<= z 3.4e-220)
         t_0
         (if (<= z 2.45e-191)
           t_1
           (if (<= z 2.2e-163)
             (+ y (* (log z) y))
             (if (or (<= z 2.55e-53) (not (<= z 3.4e-9))) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (log(z) + 1.0);
	double t_1 = (x * 0.5) - (z * y);
	double tmp;
	if (z <= 6.5e-282) {
		tmp = t_0;
	} else if (z <= 6.6e-238) {
		tmp = t_1;
	} else if (z <= 3.4e-220) {
		tmp = t_0;
	} else if (z <= 2.45e-191) {
		tmp = t_1;
	} else if (z <= 2.2e-163) {
		tmp = y + (log(z) * y);
	} else if ((z <= 2.55e-53) || !(z <= 3.4e-9)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (log(z) + 1.0d0)
    t_1 = (x * 0.5d0) - (z * y)
    if (z <= 6.5d-282) then
        tmp = t_0
    else if (z <= 6.6d-238) then
        tmp = t_1
    else if (z <= 3.4d-220) then
        tmp = t_0
    else if (z <= 2.45d-191) then
        tmp = t_1
    else if (z <= 2.2d-163) then
        tmp = y + (log(z) * y)
    else if ((z <= 2.55d-53) .or. (.not. (z <= 3.4d-9))) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (Math.log(z) + 1.0);
	double t_1 = (x * 0.5) - (z * y);
	double tmp;
	if (z <= 6.5e-282) {
		tmp = t_0;
	} else if (z <= 6.6e-238) {
		tmp = t_1;
	} else if (z <= 3.4e-220) {
		tmp = t_0;
	} else if (z <= 2.45e-191) {
		tmp = t_1;
	} else if (z <= 2.2e-163) {
		tmp = y + (Math.log(z) * y);
	} else if ((z <= 2.55e-53) || !(z <= 3.4e-9)) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (math.log(z) + 1.0)
	t_1 = (x * 0.5) - (z * y)
	tmp = 0
	if z <= 6.5e-282:
		tmp = t_0
	elif z <= 6.6e-238:
		tmp = t_1
	elif z <= 3.4e-220:
		tmp = t_0
	elif z <= 2.45e-191:
		tmp = t_1
	elif z <= 2.2e-163:
		tmp = y + (math.log(z) * y)
	elif (z <= 2.55e-53) or not (z <= 3.4e-9):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(log(z) + 1.0))
	t_1 = Float64(Float64(x * 0.5) - Float64(z * y))
	tmp = 0.0
	if (z <= 6.5e-282)
		tmp = t_0;
	elseif (z <= 6.6e-238)
		tmp = t_1;
	elseif (z <= 3.4e-220)
		tmp = t_0;
	elseif (z <= 2.45e-191)
		tmp = t_1;
	elseif (z <= 2.2e-163)
		tmp = Float64(y + Float64(log(z) * y));
	elseif ((z <= 2.55e-53) || !(z <= 3.4e-9))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (log(z) + 1.0);
	t_1 = (x * 0.5) - (z * y);
	tmp = 0.0;
	if (z <= 6.5e-282)
		tmp = t_0;
	elseif (z <= 6.6e-238)
		tmp = t_1;
	elseif (z <= 3.4e-220)
		tmp = t_0;
	elseif (z <= 2.45e-191)
		tmp = t_1;
	elseif (z <= 2.2e-163)
		tmp = y + (log(z) * y);
	elseif ((z <= 2.55e-53) || ~((z <= 3.4e-9)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[Log[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 6.5e-282], t$95$0, If[LessEqual[z, 6.6e-238], t$95$1, If[LessEqual[z, 3.4e-220], t$95$0, If[LessEqual[z, 2.45e-191], t$95$1, If[LessEqual[z, 2.2e-163], N[(y + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 2.55e-53], N[Not[LessEqual[z, 3.4e-9]], $MachinePrecision]], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 6.50000000000000012e-282 or 6.59999999999999939e-238 < z < 3.39999999999999993e-220 or 2.55000000000000022e-53 < z < 3.3999999999999998e-9

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-+l+ 99.5%

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

      3. distribute-lft-in 99.4%

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

      4. *-rgt-identity 99.4%

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

      5. associate-+r+ 99.4%

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

      6. fma-def 99.4%

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

      7. +-commutative 99.4%

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

      8. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in z around 0 98.1%

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

   5. Taylor expanded in y around inf 84.0%

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

   if 6.50000000000000012e-282 < z < 6.59999999999999939e-238 or 3.39999999999999993e-220 < z < 2.45e-191 or 2.20000000000000011e-163 < z < 2.55000000000000022e-53 or 3.3999999999999998e-9 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 88.6%

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

      1. mul-1-neg 88.6%

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

      2. distribute-rgt-neg-out 88.6%

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

   4. Simplified 88.6%

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

      1. fma-def 88.6%

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

      2. distribute-rgt-neg-out 88.6%

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

      3. add-sqr-sqrt 88.3%

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

      4. sqrt-unprod 67.3%

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

      5. sqr-neg 67.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 39.6%

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

      8. fma-neg 39.6%

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

      9. *-commutative 39.6%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 67.3%

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

      12. sqr-neg 67.3%

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

      13. sqrt-unprod 88.3%

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

      14. add-sqr-sqrt 88.6%

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

   6. Applied egg-rr 88.6%

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

   if 2.45e-191 < z < 2.20000000000000011e-163

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-+l+ 99.5%

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

      3. distribute-lft-in 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.7%

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

   5. Taylor expanded in x around 0 72.9%

      \[\leadsto \color{blue}{y \cdot \log z + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{-282}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-238}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-220}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-191}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-163}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-53} \lor \neg \left(z \leq 3.4 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \end{array} \]

Alternative 5: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{+17}:\\ \;\;\;\;y \cdot \left(\left(\log z + 1\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -5e-58)
   (fma (- z) y (* x 0.5))
   (if (<= (* x 0.5) 1e+17)
     (* y (- (+ (log z) 1.0) z))
     (- (* x 0.5) (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -5e-58) {
		tmp = fma(-z, y, (x * 0.5));
	} else if ((x * 0.5) <= 1e+17) {
		tmp = y * ((log(z) + 1.0) - z);
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -5e-58)
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	elseif (Float64(x * 0.5) <= 1e+17)
		tmp = Float64(y * Float64(Float64(log(z) + 1.0) - z));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -5e-58], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 1e+17], N[(y * N[(N[(N[Log[z], $MachinePrecision] + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x 1/2) < -4.99999999999999977e-58

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. 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. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. sub-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 90.5%

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

      1. neg-mul-1 90.5%

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

   6. Simplified 90.5%

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

   if -4.99999999999999977e-58 < (*.f64 x 1/2) < 1e17

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. 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. fma-def 99.8%

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

      4. sub-neg 99.8%

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

      5. associate-+l+ 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around inf 89.4%

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

   if 1e17 < (*.f64 x 1/2)

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 88.1%

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

      1. mul-1-neg 88.1%

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

      2. distribute-rgt-neg-out 88.1%

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

   4. Simplified 88.1%

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

      1. fma-def 88.1%

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

      2. distribute-rgt-neg-out 88.1%

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

      3. add-sqr-sqrt 87.8%

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

      4. sqrt-unprod 71.3%

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

      5. sqr-neg 71.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 52.8%

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

      8. fma-neg 52.8%

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

      9. *-commutative 52.8%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 71.3%

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

      12. sqr-neg 71.3%

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

      13. sqrt-unprod 87.8%

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

      14. add-sqr-sqrt 88.1%

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

   6. Applied egg-rr 88.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{+17}:\\ \;\;\;\;y \cdot \left(\left(\log z + 1\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 6: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{+17}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -5e-58)
   (fma (- z) y (* x 0.5))
   (if (<= (* x 0.5) 1e+17)
     (* y (+ 1.0 (- (log z) z)))
     (- (* x 0.5) (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -5e-58) {
		tmp = fma(-z, y, (x * 0.5));
	} else if ((x * 0.5) <= 1e+17) {
		tmp = y * (1.0 + (log(z) - z));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -5e-58)
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	elseif (Float64(x * 0.5) <= 1e+17)
		tmp = Float64(y * Float64(1.0 + Float64(log(z) - z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -5e-58], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 1e+17], N[(y * N[(1.0 + N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x 1/2) < -4.99999999999999977e-58

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. 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. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. sub-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 90.5%

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

      1. neg-mul-1 90.5%

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

   6. Simplified 90.5%

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

   if -4.99999999999999977e-58 < (*.f64 x 1/2) < 1e17

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-+l+ 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around -inf 89.4%

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

      1. mul-1-neg 89.4%

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

      2. distribute-rgt-neg-in 89.4%

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

      3. sub-neg 89.4%

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

      4. mul-1-neg 89.4%

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

      5. sub-neg 89.4%

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

      6. +-commutative 89.4%

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

      7. distribute-neg-in 89.4%

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

      8. remove-double-neg 89.4%

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

      9. sub-neg 89.4%

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

      10. metadata-eval 89.4%

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

      11. +-commutative 89.4%

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

   6. Simplified 89.4%

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

   if 1e17 < (*.f64 x 1/2)

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 88.1%

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

      1. mul-1-neg 88.1%

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

      2. distribute-rgt-neg-out 88.1%

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

   4. Simplified 88.1%

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

      1. fma-def 88.1%

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

      2. distribute-rgt-neg-out 88.1%

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

      3. add-sqr-sqrt 87.8%

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

      4. sqrt-unprod 71.3%

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

      5. sqr-neg 71.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 52.8%

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

      8. fma-neg 52.8%

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

      9. *-commutative 52.8%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 71.3%

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

      12. sqr-neg 71.3%

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

      13. sqrt-unprod 87.8%

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

      14. add-sqr-sqrt 88.1%

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

   6. Applied egg-rr 88.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{+17}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 7: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.013:\\ \;\;\;\;\log z \cdot y + \left(y + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.013) (+ (* (log z) y) (+ y (* x 0.5))) (fma (- z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.013) {
		tmp = (log(z) * y) + (y + (x * 0.5));
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.013)
		tmp = Float64(Float64(log(z) * y) + Float64(y + Float64(x * 0.5)));
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.013], N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] + N[(y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0129999999999999994

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 98.2%

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

   if 0.0129999999999999994 < 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. *-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 99.4%

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

      1. neg-mul-1 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.013:\\ \;\;\;\;\log z \cdot y + \left(y + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 9: 75.6% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in z around inf 75.3%

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

   1. mul-1-neg 75.3%

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

   2. distribute-rgt-neg-out 75.3%

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

4. Simplified 75.3%

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

   1. fma-def 75.3%

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

   2. distribute-rgt-neg-out 75.3%

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

   3. add-sqr-sqrt 75.0%

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

   4. sqrt-unprod 58.1%

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

   5. sqr-neg 58.1%

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

   6. sqrt-unprod 0.0%

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

   7. add-sqr-sqrt 35.4%

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

   8. fma-neg 35.4%

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

   9. *-commutative 35.4%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-unprod 58.1%

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

   12. sqr-neg 58.1%

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

   13. sqrt-unprod 75.0%

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

   14. add-sqr-sqrt 75.3%

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

6. Applied egg-rr 75.3%

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

7. Final simplification 75.3%

   \[\leadsto x \cdot 0.5 - z \cdot y \]

Alternative 10: 60.4% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 16500000000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 16500000000000.0) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 16500000000000.0) {
		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 <= 16500000000000.0d0) 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 <= 16500000000000.0) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 16500000000000.0:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 16500000000000.0)
		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 <= 16500000000000.0)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 16500000000000.0], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.65e13

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 48.1%

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

   if 1.65e13 < 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. *-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. *-commutative 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

   6. Simplified 75.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 16500000000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 11: 41.1% 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 99.8%

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

2. Taylor expanded in x around inf 36.3%

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

3. Final simplification 36.3%

   \[\leadsto x \cdot 0.5 \]

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 2023199 
(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)))))