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

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

Alternatives: 7

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 99.8%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
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. 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: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + y \cdot \log z\\ \mathbf{if}\;z \leq 1.9 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-101}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (* y (log z)))))
   (if (<= z 1.9e-128)
     t_0
     (if (<= z 1.1e-101)
       (- (* x 0.5) (* y z))
       (if (<= z 1.6e-53) 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 tmp;
	if (z <= 1.9e-128) {
		tmp = t_0;
	} else if (z <= 1.1e-101) {
		tmp = (x * 0.5) - (y * z);
	} else if (z <= 1.6e-53) {
		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)))
	tmp = 0.0
	if (z <= 1.9e-128)
		tmp = t_0;
	elseif (z <= 1.1e-101)
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	elseif (z <= 1.6e-53)
		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]}, If[LessEqual[z, 1.9e-128], t$95$0, If[LessEqual[z, 1.1e-101], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e-53], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.9000000000000001e-128 or 1.0999999999999999e-101 < z < 1.6e-53

   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. distribute-lft-in 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 60.2%

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

   5. Taylor expanded in z around 0 60.2%

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

   if 1.9000000000000001e-128 < z < 1.0999999999999999e-101

   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%

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

      2. add-cube-cbrt 99.3%

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

      3. fma-def 99.3%

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

      4. pow2 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-udef 99.3%

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

      2. unpow2 99.3%

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

      3. add-cube-cbrt 99.9%

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

      4. distribute-lft-in 99.8%

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

      5. +-commutative 99.8%

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

      6. flip-+ 94.4%

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

      7. clear-num 94.4%

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

      8. clear-num 94.3%

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

      9. flip-+ 99.7%

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

      10. distribute-lft-in 99.8%

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

      11. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around inf 73.8%

      \[\leadsto x \cdot 0.5 + \frac{1}{\color{blue}{\frac{-1}{y \cdot z}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 73.8%

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

      2. metadata-eval 73.8%

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

      3. mul-1-neg 73.8%

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

      4. unsub-neg 73.8%

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

   8. Applied egg-rr 73.8%

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

   if 1.6e-53 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-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 95.6%

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

      1. mul-1-neg 95.6%

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

   6. Simplified 95.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{-128}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-101}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-53}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-129} \lor \neg \left(z \leq 2.8 \cdot 10^{-101}\right) \land z \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 2.4e-129) (and (not (<= z 2.8e-101)) (<= z 1.05e-47)))
   (+ y (* y (log z)))
   (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.4e-129) || (!(z <= 2.8e-101) && (z <= 1.05e-47))) {
		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 <= 2.4d-129) .or. (.not. (z <= 2.8d-101)) .and. (z <= 1.05d-47)) 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 <= 2.4e-129) || (!(z <= 2.8e-101) && (z <= 1.05e-47))) {
		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 <= 2.4e-129) or (not (z <= 2.8e-101) and (z <= 1.05e-47)):
		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 <= 2.4e-129) || (!(z <= 2.8e-101) && (z <= 1.05e-47)))
		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 <= 2.4e-129) || (~((z <= 2.8e-101)) && (z <= 1.05e-47)))
		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, 2.4e-129], And[N[Not[LessEqual[z, 2.8e-101]], $MachinePrecision], LessEqual[z, 1.05e-47]]], 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 < 2.39999999999999989e-129 or 2.79999999999999989e-101 < z < 1.05e-47

   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. distribute-lft-in 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 60.2%

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

   5. Taylor expanded in z around 0 60.2%

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

   if 2.39999999999999989e-129 < z < 2.79999999999999989e-101 or 1.05e-47 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. add-cube-cbrt 99.8%

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

      3. fma-def 99.8%

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

      4. pow2 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. unpow2 99.8%

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

      3. add-cube-cbrt 99.9%

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

      4. distribute-lft-in 99.3%

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

      5. +-commutative 99.3%

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

      6. flip-+ 58.6%

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

      7. clear-num 58.6%

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

      8. clear-num 58.6%

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

      9. flip-+ 99.2%

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

      10. distribute-lft-in 99.9%

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

      11. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around inf 93.1%

      \[\leadsto x \cdot 0.5 + \frac{1}{\color{blue}{\frac{-1}{y \cdot z}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 93.1%

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

      2. metadata-eval 93.1%

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

      3. mul-1-neg 93.1%

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

      4. unsub-neg 93.1%

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

   8. Applied egg-rr 93.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-129} \lor \neg \left(z \leq 2.8 \cdot 10^{-101}\right) \land z \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.28)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.28], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 98.5%

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

   if 0.28000000000000003 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-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 99.2%

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

      1. mul-1-neg 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 98.8%

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

Alternative 5: 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 99.8%

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

2. Final simplification 99.8%

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

Alternative 6: 74.6% 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 99.8%

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

   1. +-commutative 99.8%

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

   2. add-cube-cbrt 99.5%

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

   3. fma-def 99.5%

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

   4. pow2 99.5%

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

3. Applied egg-rr 99.5%

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

   1. fma-udef 99.5%

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

   2. unpow2 99.5%

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

   3. add-cube-cbrt 99.8%

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

   4. distribute-lft-in 99.4%

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

   5. +-commutative 99.4%

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

   6. flip-+ 63.0%

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

   7. clear-num 62.9%

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

   8. clear-num 63.0%

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

   9. flip-+ 99.4%

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

   10. distribute-lft-in 99.8%

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

   11. +-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in z around inf 73.8%

   \[\leadsto x \cdot 0.5 + \frac{1}{\color{blue}{\frac{-1}{y \cdot z}}} \]
7. Step-by-step derivation

   1. associate-/r/ 73.8%

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

   2. metadata-eval 73.8%

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

   3. mul-1-neg 73.8%

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

   4. unsub-neg 73.8%

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

8. Applied egg-rr 73.8%

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

9. Final simplification 73.8%

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

Alternative 7: 40.8% 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 40.4%

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

3. Final simplification 40.4%

   \[\leadsto x \cdot 0.5 \]

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 2023340 
(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)))))