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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 8

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, 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 2: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-73} \lor \neg \left(x \cdot 0.5 \leq 4 \cdot 10^{-110}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -2e-73) (not (<= (* x 0.5) 4e-110)))
   (- (* x 0.5) (* y z))
   (* y (- (+ 1.0 (log z)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -2e-73) || !((x * 0.5) <= 4e-110)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * ((1.0 + log(z)) - 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 (((x * 0.5d0) <= (-2d-73)) .or. (.not. ((x * 0.5d0) <= 4d-110))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * ((1.0d0 + log(z)) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -2e-73) || !((x * 0.5) <= 4e-110)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * ((1.0 + Math.log(z)) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -2e-73) or not ((x * 0.5) <= 4e-110):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * ((1.0 + math.log(z)) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -2e-73) || !(Float64(x * 0.5) <= 4e-110))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -2e-73) || ~(((x * 0.5) <= 4e-110)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * ((1.0 + log(z)) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -2e-73], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 4e-110]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -1.99999999999999999e-73 or 4.0000000000000002e-110 < (*.f64 x 1/2)

   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 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-rgt-neg-out 87.1%

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

   4. Simplified 87.1%

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

      1. fma-def 87.1%

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

      2. distribute-rgt-neg-out 87.1%

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

      3. add-sqr-sqrt 87.1%

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

      4. sqrt-unprod 67.2%

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

      5. sqr-neg 67.2%

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

      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 60.4%

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

      8. fma-neg 60.4%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 67.2%

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

      11. sqr-neg 67.2%

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

      12. sqrt-unprod 87.1%

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

      13. add-sqr-sqrt 87.1%

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

   6. Applied egg-rr 87.1%

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

   if -1.99999999999999999e-73 < (*.f64 x 1/2) < 4.0000000000000002e-110

   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 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. associate-+r+ 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around -inf 91.7%

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

      1. mul-1-neg 91.7%

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

      2. distribute-rgt-neg-in 91.7%

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

      3. sub-neg 91.7%

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

      4. mul-1-neg 91.7%

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

      5. sub-neg 91.7%

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

      6. +-commutative 91.7%

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

      7. distribute-neg-in 91.7%

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

      8. remove-double-neg 91.7%

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

      9. sub-neg 91.7%

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

      10. metadata-eval 91.7%

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

      11. +-commutative 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in y around 0 91.7%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-73} \lor \neg \left(x \cdot 0.5 \leq 4 \cdot 10^{-110}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \end{array} \]

Alternative 3: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-220} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-282}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -1e-220) (not (<= (* x 0.5) 2e-282)))
   (- (* x 0.5) (* y z))
   (* y (+ 1.0 (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -1e-220) || !((x * 0.5) <= 2e-282)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + log(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 (((x * 0.5d0) <= (-1d-220)) .or. (.not. ((x * 0.5d0) <= 2d-282))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * (1.0d0 + log(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -1e-220) || !((x * 0.5) <= 2e-282)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -1e-220) or not ((x * 0.5) <= 2e-282):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * (1.0 + math.log(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -1e-220) || !(Float64(x * 0.5) <= 2e-282))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(1.0 + log(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -1e-220) || ~(((x * 0.5) <= 2e-282)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * (1.0 + log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-220], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-282]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -9.99999999999999992e-221 or 2e-282 < (*.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 79.6%

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

      1. mul-1-neg 79.6%

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

      2. distribute-rgt-neg-out 79.6%

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

   4. Simplified 79.6%

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

      1. fma-def 79.6%

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

      2. distribute-rgt-neg-out 79.6%

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

      3. add-sqr-sqrt 79.4%

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

      4. sqrt-unprod 59.7%

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

      5. sqr-neg 59.7%

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

      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 45.5%

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

      8. fma-neg 45.5%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 59.7%

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

      11. sqr-neg 59.7%

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

      12. sqrt-unprod 79.4%

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

      13. add-sqr-sqrt 79.6%

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

   6. Applied egg-rr 79.6%

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

   if -9.99999999999999992e-221 < (*.f64 x 1/2) < 2e-282

   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. sub-neg 99.6%

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

      2. associate-+l+ 99.6%

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

      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 91.9%

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

      1. mul-1-neg 91.9%

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

      2. distribute-rgt-neg-in 91.9%

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

      3. sub-neg 91.9%

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

      4. mul-1-neg 91.9%

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

      5. sub-neg 91.9%

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

      6. +-commutative 91.9%

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

      7. distribute-neg-in 91.9%

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

      8. remove-double-neg 91.9%

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

      9. sub-neg 91.9%

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

      10. metadata-eval 91.9%

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

      11. +-commutative 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in z around 0 67.3%

      \[\leadsto \color{blue}{\left(1 + \log z\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-220} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-282}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \]

Alternative 4: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-220} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-282}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -1e-220) (not (<= (* x 0.5) 2e-282)))
   (- (* x 0.5) (* y z))
   (+ y (* y (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -1e-220) || !((x * 0.5) <= 2e-282)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y + (y * log(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 (((x * 0.5d0) <= (-1d-220)) .or. (.not. ((x * 0.5d0) <= 2d-282))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y + (y * log(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -1e-220) || !((x * 0.5) <= 2e-282)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y + (y * Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -1e-220) or not ((x * 0.5) <= 2e-282):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y + (y * math.log(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -1e-220) || !(Float64(x * 0.5) <= 2e-282))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y + Float64(y * log(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -1e-220) || ~(((x * 0.5) <= 2e-282)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y + (y * log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-220], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-282]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -9.99999999999999992e-221 or 2e-282 < (*.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 79.6%

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

      1. mul-1-neg 79.6%

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

      2. distribute-rgt-neg-out 79.6%

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

   4. Simplified 79.6%

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

      1. fma-def 79.6%

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

      2. distribute-rgt-neg-out 79.6%

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

      3. add-sqr-sqrt 79.4%

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

      4. sqrt-unprod 59.7%

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

      5. sqr-neg 59.7%

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

      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 45.5%

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

      8. fma-neg 45.5%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 59.7%

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

      11. sqr-neg 59.7%

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

      12. sqrt-unprod 79.4%

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

      13. add-sqr-sqrt 79.6%

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

   6. Applied egg-rr 79.6%

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

   if -9.99999999999999992e-221 < (*.f64 x 1/2) < 2e-282

   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. sub-neg 99.6%

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

      2. associate-+l+ 99.6%

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

      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 91.9%

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

      1. mul-1-neg 91.9%

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

      2. distribute-rgt-neg-in 91.9%

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

      3. sub-neg 91.9%

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

      4. mul-1-neg 91.9%

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

      5. sub-neg 91.9%

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

      6. +-commutative 91.9%

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

      7. distribute-neg-in 91.9%

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

      8. remove-double-neg 91.9%

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

      9. sub-neg 91.9%

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

      10. metadata-eval 91.9%

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

      11. +-commutative 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in z around 0 67.3%

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

      1. +-commutative 67.3%

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

      2. distribute-rgt1-in 67.4%

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

   9. Simplified 67.4%

      \[\leadsto \color{blue}{y + \log z \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-220} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-282}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \]

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.00024:\\ \;\;\;\;x \cdot 0.5 + \left(y + y \cdot \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.00024) (+ (* x 0.5) (+ y (* y (log z)))) (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.00024) {
		tmp = (x * 0.5) + (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 <= 0.00024d0) then
        tmp = (x * 0.5d0) + (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 <= 0.00024) {
		tmp = (x * 0.5) + (y + (y * Math.log(z)));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.00024:
		tmp = (x * 0.5) + (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 <= 0.00024)
		tmp = Float64(Float64(x * 0.5) + 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 <= 0.00024)
		tmp = (x * 0.5) + (y + (y * log(z)));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.00024], N[(N[(x * 0.5), $MachinePrecision] + N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.40000000000000006e-4

   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 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. *-rgt-identity 99.7%

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

   4. Simplified 99.7%

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

   if 2.40000000000000006e-4 < 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 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

   4. Simplified 98.1%

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

      1. fma-def 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

      3. add-sqr-sqrt 97.8%

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

      4. sqrt-unprod 61.4%

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

      5. sqr-neg 61.4%

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

      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 32.5%

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

      8. fma-neg 32.5%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 61.4%

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

      11. sqr-neg 61.4%

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

      12. sqrt-unprod 97.8%

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

      13. add-sqr-sqrt 98.1%

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

   6. Applied egg-rr 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.00024:\\ \;\;\;\;x \cdot 0.5 + \left(y + y \cdot \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 6: 75.5% 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 75.4%

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

   1. mul-1-neg 75.4%

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

   2. distribute-rgt-neg-out 75.4%

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

4. Simplified 75.4%

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

   1. fma-def 75.4%

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

   2. distribute-rgt-neg-out 75.4%

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

   3. add-sqr-sqrt 75.3%

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

   4. sqrt-unprod 57.0%

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

   5. sqr-neg 57.0%

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

   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 42.1%

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

   8. fma-neg 42.1%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 57.0%

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

   11. sqr-neg 57.0%

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

   12. sqrt-unprod 75.3%

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

   13. add-sqr-sqrt 75.4%

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

6. Applied egg-rr 75.4%

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

7. Final simplification 75.4%

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

Alternative 7: 60.4% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+29}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.5e+29) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.5e+29) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.5d+29) then
        tmp = x * 0.5d0
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.5e+29) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.5e+29:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.5e+29)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.5e+29)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.5e+29], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.5e29

   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 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. associate-+r+ 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 52.8%

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

   if 2.5e29 < 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. sub-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-rgt-identity 100.0%

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

      5. associate-+r+ 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around -inf 71.9%

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

      1. mul-1-neg 71.9%

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

      2. distribute-rgt-neg-in 71.9%

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

      3. sub-neg 71.9%

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

      4. mul-1-neg 71.9%

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

      5. sub-neg 71.9%

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

      6. +-commutative 71.9%

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

      7. distribute-neg-in 71.9%

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

      8. remove-double-neg 71.9%

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

      9. sub-neg 71.9%

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

      10. metadata-eval 71.9%

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

      11. +-commutative 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in z around inf 71.9%

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

      1. mul-1-neg 71.9%

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

      2. distribute-rgt-neg-out 71.9%

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

   9. Simplified 71.9%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+29}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 8: 40.4% 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. Step-by-step derivation

   1. sub-neg 99.9%

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

   2. associate-+l+ 99.9%

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

   3. distribute-lft-in 99.9%

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

   4. *-rgt-identity 99.9%

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

   5. associate-+r+ 99.9%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. unsub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 43.1%

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

5. Final simplification 43.1%

   \[\leadsto x \cdot 0.5 \]

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 2023221 
(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)))))