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

Percentage Accurate: 99.9% → 99.9%

Time: 8.1s

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-def 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. Final simplification 99.9%

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

Alternative 2: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + y \cdot \log z\\ t_1 := x \cdot 0.5 - y \cdot z\\ \mathbf{if}\;z \leq 1.65 \cdot 10^{-250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (* y (log z)))) (t_1 (- (* x 0.5) (* y z))))
   (if (<= z 1.65e-250)
     t_0
     (if (<= z 1.05e-134)
       t_1
       (if (<= z 5e-75)
         t_0
         (if (<= z 1.1e-18)
           t_1
           (if (<= z 1.35e-6) t_0 (fma y (- z) (* x 0.5)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (y * log(z));
	double t_1 = (x * 0.5) - (y * z);
	double tmp;
	if (z <= 1.65e-250) {
		tmp = t_0;
	} else if (z <= 1.05e-134) {
		tmp = t_1;
	} else if (z <= 5e-75) {
		tmp = t_0;
	} else if (z <= 1.1e-18) {
		tmp = t_1;
	} else if (z <= 1.35e-6) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(y * log(z)))
	t_1 = Float64(Float64(x * 0.5) - Float64(y * z))
	tmp = 0.0
	if (z <= 1.65e-250)
		tmp = t_0;
	elseif (z <= 1.05e-134)
		tmp = t_1;
	elseif (z <= 5e-75)
		tmp = t_0;
	elseif (z <= 1.1e-18)
		tmp = t_1;
	elseif (z <= 1.35e-6)
		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[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.65e-250], t$95$0, If[LessEqual[z, 1.05e-134], t$95$1, If[LessEqual[z, 5e-75], t$95$0, If[LessEqual[z, 1.1e-18], t$95$1, If[LessEqual[z, 1.35e-6], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.65e-250 or 1.05e-134 < z < 4.99999999999999979e-75 or 1.0999999999999999e-18 < z < 1.34999999999999999e-6

   1. Initial program 99.6%

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

      1. distribute-lft-in 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 97.7%

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

   5. Taylor expanded in x around 0 71.0%

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

   if 1.65e-250 < z < 1.05e-134 or 4.99999999999999979e-75 < z < 1.0999999999999999e-18

   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 71.0%

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

      1. mul-1-neg 71.0%

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

      2. *-commutative 71.0%

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

      3. distribute-rgt-neg-in 71.0%

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

   4. Simplified 71.0%

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

      1. fma-def 71.0%

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

      2. distribute-rgt-neg-out 71.0%

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

      3. add-sqr-sqrt 45.4%

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

      4. sqrt-unprod 63.4%

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

      5. sqr-neg 63.4%

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

      6. sqrt-unprod 25.0%

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

      7. add-sqr-sqrt 70.2%

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

      8. fma-neg 70.2%

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

      9. *-commutative 70.2%

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

      10. add-sqr-sqrt 25.0%

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

      11. sqrt-unprod 63.4%

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

      12. sqr-neg 63.4%

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

      13. sqrt-unprod 45.4%

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

      14. add-sqr-sqrt 71.0%

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

   6. Applied egg-rr 71.0%

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

   if 1.34999999999999999e-6 < 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-def 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. Taylor expanded in z around inf 98.9%

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

      1. mul-1-neg 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{-250}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-75}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-250} \lor \neg \left(z \leq 1.2 \cdot 10^{-134}\right) \land \left(z \leq 9.4 \cdot 10^{-77} \lor \neg \left(z \leq 1.7 \cdot 10^{-18}\right) \land z \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 2.7e-250)
         (and (not (<= z 1.2e-134))
              (or (<= z 9.4e-77) (and (not (<= z 1.7e-18)) (<= z 2.2e-6)))))
   (+ y (* y (log z)))
   (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.7e-250) || (!(z <= 1.2e-134) && ((z <= 9.4e-77) || (!(z <= 1.7e-18) && (z <= 2.2e-6))))) {
		tmp = y + (y * 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 <= 2.7d-250) .or. (.not. (z <= 1.2d-134)) .and. (z <= 9.4d-77) .or. (.not. (z <= 1.7d-18)) .and. (z <= 2.2d-6)) then
        tmp = y + (y * 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 <= 2.7e-250) || (!(z <= 1.2e-134) && ((z <= 9.4e-77) || (!(z <= 1.7e-18) && (z <= 2.2e-6))))) {
		tmp = y + (y * Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 2.7e-250) or (not (z <= 1.2e-134) and ((z <= 9.4e-77) or (not (z <= 1.7e-18) and (z <= 2.2e-6)))):
		tmp = y + (y * math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 2.7e-250) || (!(z <= 1.2e-134) && ((z <= 9.4e-77) || (!(z <= 1.7e-18) && (z <= 2.2e-6)))))
		tmp = Float64(y + Float64(y * 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 <= 2.7e-250) || (~((z <= 1.2e-134)) && ((z <= 9.4e-77) || (~((z <= 1.7e-18)) && (z <= 2.2e-6)))))
		tmp = y + (y * log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 2.7e-250], And[N[Not[LessEqual[z, 1.2e-134]], $MachinePrecision], Or[LessEqual[z, 9.4e-77], And[N[Not[LessEqual[z, 1.7e-18]], $MachinePrecision], LessEqual[z, 2.2e-6]]]]], N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.70000000000000002e-250 or 1.20000000000000005e-134 < z < 9.3999999999999998e-77 or 1.70000000000000001e-18 < z < 2.2000000000000001e-6

   1. Initial program 99.6%

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

      1. distribute-lft-in 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around 0 97.7%

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

   5. Taylor expanded in x around 0 71.0%

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

   if 2.70000000000000002e-250 < z < 1.20000000000000005e-134 or 9.3999999999999998e-77 < z < 1.70000000000000001e-18 or 2.2000000000000001e-6 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. *-commutative 88.7%

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

      3. distribute-rgt-neg-in 88.7%

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

   4. Simplified 88.7%

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

      1. fma-def 88.7%

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

      2. distribute-rgt-neg-out 88.7%

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

      3. add-sqr-sqrt 48.6%

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

      4. sqrt-unprod 57.6%

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

      5. sqr-neg 57.6%

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

      6. sqrt-unprod 16.0%

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

      7. add-sqr-sqrt 40.5%

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

      8. fma-neg 40.5%

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

      9. *-commutative 40.5%

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

      10. add-sqr-sqrt 16.0%

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

      11. sqrt-unprod 57.6%

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

      12. sqr-neg 57.6%

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

      13. sqrt-unprod 48.6%

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

      14. add-sqr-sqrt 88.7%

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

   6. Applied egg-rr 88.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-250} \lor \neg \left(z \leq 1.2 \cdot 10^{-134}\right) \land \left(z \leq 9.4 \cdot 10^{-77} \lor \neg \left(z \leq 1.7 \cdot 10^{-18}\right) \land z \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-62}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;x \cdot 0.5 \leq 0.1:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -1e-62)
   (- (* x 0.5) (* y z))
   (if (<= (* x 0.5) 0.1)
     (* y (+ 1.0 (- (log z) z)))
     (fma y (- z) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -1e-62) {
		tmp = (x * 0.5) - (y * z);
	} else if ((x * 0.5) <= 0.1) {
		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 (Float64(x * 0.5) <= -1e-62)
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	elseif (Float64(x * 0.5) <= 0.1)
		tmp = Float64(y * Float64(1.0 + Float64(log(z) - z)));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-62], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 0.1], N[(y * N[(1.0 + N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x 1/2) < -1e-62

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 92.8%

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

      1. mul-1-neg 92.8%

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

      2. *-commutative 92.8%

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

      3. distribute-rgt-neg-in 92.8%

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

   4. Simplified 92.8%

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

      1. fma-def 92.8%

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

      2. distribute-rgt-neg-out 92.8%

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

      3. add-sqr-sqrt 45.3%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 24.7%

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

      7. add-sqr-sqrt 54.7%

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

      8. fma-neg 54.7%

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

      9. *-commutative 54.7%

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

      10. add-sqr-sqrt 24.7%

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

      11. sqrt-unprod 62.6%

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

      12. sqr-neg 62.6%

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

      13. sqrt-unprod 45.3%

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

      14. add-sqr-sqrt 92.8%

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

   6. Applied egg-rr 92.8%

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

   if -1e-62 < (*.f64 x 1/2) < 0.10000000000000001

   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. add-cube-cbrt 98.4%

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

      4. associate-*r* 98.3%

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

      5. fma-def 98.3%

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

      6. pow2 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in x around 0 91.2%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(y \cdot \left(\left(1 + \log z\right) - z\right)\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 91.2%

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

      2. *-lft-identity 91.2%

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

      3. associate--l+ 91.2%

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

   6. Simplified 91.2%

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

   if 0.10000000000000001 < (*.f64 x 1/2)

   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-def 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. Taylor expanded in z around inf 91.6%

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

      1. mul-1-neg 91.6%

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

   6. Simplified 91.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-62}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;x \cdot 0.5 \leq 0.1:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(2 + \left(\log z + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.8e-6)
   (+ (* x 0.5) (* y (+ 2.0 (+ (log z) -1.0))))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.8e-6) {
		tmp = (x * 0.5) + (y * (2.0 + (log(z) + -1.0)));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.8e-6)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(2.0 + Float64(log(z) + -1.0))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.8e-6], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(2.0 + N[(N[Log[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8000000000000004e-6

   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%

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

      2. flip-+ 99.7%

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

      3. associate-*l/ 99.7%

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

      4. pow2 99.7%

         \[\leadsto x \cdot 0.5 + \frac{\left(\color{blue}{{\left(1 - z\right)}^{2}} - \log z \cdot \log z\right) \cdot y}{\left(1 - z\right) - \log z} \]

      5. pow2 99.7%

         \[\leadsto x \cdot 0.5 + \frac{\left({\left(1 - z\right)}^{2} - \color{blue}{{\log z}^{2}}\right) \cdot y}{\left(1 - z\right) - \log z} \]

      6. associate--l- 99.7%

         \[\leadsto x \cdot 0.5 + \frac{\left({\left(1 - z\right)}^{2} - {\log z}^{2}\right) \cdot y}{\color{blue}{1 - \left(z + \log z\right)}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto x \cdot 0.5 + \color{blue}{\frac{\left({\left(1 - z\right)}^{2} - {\log z}^{2}\right) \cdot y}{1 - \left(z + \log z\right)}} \]

   4. Taylor expanded in z around inf 99.7%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(\left(-1 \cdot \left(y \cdot z\right) + 2 \cdot y\right) - y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{z}\right)\right)\right)} \]

   5. Taylor expanded in z around 0 98.8%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(2 \cdot y - y \cdot \left(1 - \log z\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto x \cdot 0.5 + \left(\color{blue}{y \cdot 2} - y \cdot \left(1 - \log z\right)\right) \]

      2. distribute-lft-out-- 98.8%

         \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(2 - \left(1 - \log z\right)\right)} \]

      3. remove-double-neg 98.8%

         \[\leadsto x \cdot 0.5 + y \cdot \left(2 - \left(1 - \color{blue}{\left(-\left(-\log z\right)\right)}\right)\right) \]

      4. log-rec 98.8%

         \[\leadsto x \cdot 0.5 + y \cdot \left(2 - \left(1 - \left(-\color{blue}{\log \left(\frac{1}{z}\right)}\right)\right)\right) \]

      5. mul-1-neg 98.8%

         \[\leadsto x \cdot 0.5 + y \cdot \left(2 - \left(1 - \color{blue}{-1 \cdot \log \left(\frac{1}{z}\right)}\right)\right) \]

      6. unsub-neg 98.8%

         \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(2 + \left(-\left(1 - -1 \cdot \log \left(\frac{1}{z}\right)\right)\right)\right)} \]

      7. cancel-sign-sub-inv 98.8%

         \[\leadsto x \cdot 0.5 + y \cdot \left(2 + \left(-\color{blue}{\left(1 + \left(--1\right) \cdot \log \left(\frac{1}{z}\right)\right)}\right)\right) \]

      8. metadata-eval 98.8%

         \[\leadsto x \cdot 0.5 + y \cdot \left(2 + \left(-\left(1 + \color{blue}{1} \cdot \log \left(\frac{1}{z}\right)\right)\right)\right) \]

      9. *-lft-identity 98.8%

         \[\leadsto x \cdot 0.5 + y \cdot \left(2 + \left(-\left(1 + \color{blue}{\log \left(\frac{1}{z}\right)}\right)\right)\right) \]

      10. distribute-neg-in 98.8%

          \[\leadsto x \cdot 0.5 + y \cdot \left(2 + \color{blue}{\left(\left(-1\right) + \left(-\log \left(\frac{1}{z}\right)\right)\right)}\right) \]

      11. metadata-eval 98.8%

          \[\leadsto x \cdot 0.5 + y \cdot \left(2 + \left(\color{blue}{-1} + \left(-\log \left(\frac{1}{z}\right)\right)\right)\right) \]

      12. log-rec 98.8%

          \[\leadsto x \cdot 0.5 + y \cdot \left(2 + \left(-1 + \left(-\color{blue}{\left(-\log z\right)}\right)\right)\right) \]

      13. remove-double-neg 98.8%

          \[\leadsto x \cdot 0.5 + y \cdot \left(2 + \left(-1 + \color{blue}{\log z}\right)\right) \]

   7. Simplified 98.8%

      \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(2 + \left(-1 + \log z\right)\right)} \]

   if 5.8000000000000004e-6 < 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-def 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. Taylor expanded in z around inf 98.9%

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

      1. mul-1-neg 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(2 + \left(\log z + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.8e-6)
   (+ (* 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 <= 5.8e-6) {
		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 <= 5.8e-6)
		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, 5.8e-6], 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 < 5.8000000000000004e-6

   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 0 98.8%

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

   if 5.8000000000000004e-6 < 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-def 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. Taylor expanded in z around inf 98.9%

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

      1. mul-1-neg 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 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. Final simplification 99.9%

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

Alternative 8: 75.1% 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. Taylor expanded in z around inf 73.8%

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

   1. mul-1-neg 73.8%

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

   2. *-commutative 73.8%

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

   3. distribute-rgt-neg-in 73.8%

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

4. Simplified 73.8%

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

   1. fma-def 73.8%

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

   2. distribute-rgt-neg-out 73.8%

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

   3. add-sqr-sqrt 39.9%

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

   4. sqrt-unprod 50.0%

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

   5. sqr-neg 50.0%

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

   6. sqrt-unprod 15.7%

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

   7. add-sqr-sqrt 37.4%

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

   8. fma-neg 37.4%

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

   9. *-commutative 37.4%

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

   10. add-sqr-sqrt 15.7%

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

   11. sqrt-unprod 50.0%

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

   12. sqr-neg 50.0%

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

   13. sqrt-unprod 39.9%

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

   14. add-sqr-sqrt 73.8%

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

6. Applied egg-rr 73.8%

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

7. Final simplification 73.8%

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

Alternative 9: 60.0% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2650000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2650000000.0) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2650000000.0) {
		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 <= 2650000000.0d0) 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 <= 2650000000.0) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2650000000.0:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2650000000.0)
		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 <= 2650000000.0)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2650000000.0], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.65e9

   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 51.1%

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

   if 2.65e9 < 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. add-cube-cbrt 98.9%

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

      4. associate-*r* 98.9%

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

      5. fma-def 98.9%

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

      6. pow2 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in z around inf 76.8%

      \[\leadsto \color{blue}{-1 \cdot \left({1}^{0.3333333333333333} \cdot \left(y \cdot z\right)\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 76.8%

         \[\leadsto \color{blue}{-{1}^{0.3333333333333333} \cdot \left(y \cdot z\right)} \]

      2. pow-base-1 76.8%

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

      3. *-lft-identity 76.8%

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

      4. distribute-rgt-neg-in 76.8%

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

   6. Simplified 76.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2650000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 10: 40.9% 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. Taylor expanded in x around inf 38.6%

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

3. Final simplification 38.6%

   \[\leadsto x \cdot 0.5 \]

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 2023326 
(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)))))