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

Percentage Accurate: 99.9% → 99.9%

Time: 10.6s

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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ \mathbf{if}\;z \leq 3.3 \cdot 10^{-220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-199}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-74}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))))
   (if (<= z 3.3e-220)
     t_0
     (if (<= z 3e-199)
       (- (* x 0.5) (* y z))
       (if (<= z 9.8e-74) t_0 (fma y (- z) (* x 0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double tmp;
	if (z <= 3.3e-220) {
		tmp = t_0;
	} else if (z <= 3e-199) {
		tmp = (x * 0.5) - (y * z);
	} else if (z <= 9.8e-74) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	tmp = 0.0
	if (z <= 3.3e-220)
		tmp = t_0;
	elseif (z <= 3e-199)
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	elseif (z <= 9.8e-74)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.3e-220], t$95$0, If[LessEqual[z, 3e-199], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.8e-74], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.29999999999999999e-220 or 2.99999999999999983e-199 < z < 9.8000000000000006e-74

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 55.7%

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

   4. Taylor expanded in x around 0 42.6%

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

      1. +-commutative 42.6%

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

      2. associate-/l* 41.4%

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

      3. +-commutative 41.4%

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

      4. log-rec 41.4%

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

      5. fma-undefine 41.4%

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

      6. *-commutative 41.4%

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

      7. distribute-lft-out 41.4%

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

      8. fma-undefine 41.4%

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

      9. distribute-rgt-neg-in 41.4%

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

      10. neg-mul-1 41.4%

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

      11. remove-double-neg 41.4%

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

      12. +-commutative 41.4%

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

   6. Simplified 41.4%

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

   7. Taylor expanded in z around 0 66.2%

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

   if 3.29999999999999999e-220 < z < 2.99999999999999983e-199

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 78.4%

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

      1. associate-*r* 78.4%

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

      2. mul-1-neg 78.4%

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

   5. Simplified 78.4%

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

      1. fma-define 78.4%

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

      2. distribute-lft-neg-out 78.4%

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

      3. fmm-undef 78.4%

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

   7. Applied egg-rr 78.4%

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

   if 9.8000000000000006e-74 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 93.0%

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

      1. mul-1-neg 93.0%

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

   7. Simplified 93.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{-220}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-199}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-74}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{+50} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -1e+50) (not (<= (* x 0.5) 2e-91)))
   (- (* 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) <= -1e+50) || !((x * 0.5) <= 2e-91)) {
		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) <= (-1d+50)) .or. (.not. ((x * 0.5d0) <= 2d-91))) 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) <= -1e+50) || !((x * 0.5) <= 2e-91)) {
		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) <= -1e+50) or not ((x * 0.5) <= 2e-91):
		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) <= -1e+50) || !(Float64(x * 0.5) <= 2e-91))
		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) <= -1e+50) || ~(((x * 0.5) <= 2e-91)))
		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], -1e+50], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-91]], $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 #s(literal 1/2 binary64)) < -1.0000000000000001e50 or 2.00000000000000004e-91 < (*.f64 x #s(literal 1/2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 86.1%

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

      1. associate-*r* 86.1%

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

      2. mul-1-neg 86.1%

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

   5. Simplified 86.1%

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

      1. fma-define 86.1%

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

      2. distribute-lft-neg-out 86.1%

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

      3. fmm-undef 86.1%

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

   7. Applied egg-rr 86.1%

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

   if -1.0000000000000001e50 < (*.f64 x #s(literal 1/2 binary64)) < 2.00000000000000004e-91

   1. Initial program 99.8%

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

      1. distribute-lft-in 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 88.1%

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

   6. Taylor expanded in y around 0 88.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{+50} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-220} \lor \neg \left(z \leq 6.5 \cdot 10^{-199}\right) \land z \leq 9.8 \cdot 10^{-74}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 2.9e-220) (and (not (<= z 6.5e-199)) (<= z 9.8e-74)))
   (* y (+ 1.0 (log z)))
   (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.9e-220) || (!(z <= 6.5e-199) && (z <= 9.8e-74))) {
		tmp = y * (1.0 + log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= 2.9d-220) .or. (.not. (z <= 6.5d-199)) .and. (z <= 9.8d-74)) then
        tmp = y * (1.0d0 + log(z))
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.9e-220) || (!(z <= 6.5e-199) && (z <= 9.8e-74))) {
		tmp = y * (1.0 + Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 2.9e-220) or (not (z <= 6.5e-199) and (z <= 9.8e-74)):
		tmp = y * (1.0 + math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 2.9e-220) || (!(z <= 6.5e-199) && (z <= 9.8e-74)))
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 2.9e-220) || (~((z <= 6.5e-199)) && (z <= 9.8e-74)))
		tmp = y * (1.0 + log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 2.9e-220], And[N[Not[LessEqual[z, 6.5e-199]], $MachinePrecision], LessEqual[z, 9.8e-74]]], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.8999999999999998e-220 or 6.50000000000000017e-199 < z < 9.8000000000000006e-74

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 55.7%

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

   4. Taylor expanded in x around 0 42.6%

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

      1. +-commutative 42.6%

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

      2. associate-/l* 41.4%

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

      3. +-commutative 41.4%

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

      4. log-rec 41.4%

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

      5. fma-undefine 41.4%

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

      6. *-commutative 41.4%

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

      7. distribute-lft-out 41.4%

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

      8. fma-undefine 41.4%

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

      9. distribute-rgt-neg-in 41.4%

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

      10. neg-mul-1 41.4%

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

      11. remove-double-neg 41.4%

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

      12. +-commutative 41.4%

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

   6. Simplified 41.4%

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

   7. Taylor expanded in z around 0 66.2%

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

   if 2.8999999999999998e-220 < z < 6.50000000000000017e-199 or 9.8000000000000006e-74 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 91.8%

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

      1. associate-*r* 91.8%

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

      2. mul-1-neg 91.8%

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

   5. Simplified 91.8%

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

      1. fma-define 91.8%

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

      2. distribute-lft-neg-out 91.8%

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

      3. fmm-undef 91.8%

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

   7. Applied egg-rr 91.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-220} \lor \neg \left(z \leq 6.5 \cdot 10^{-199}\right) \land z \leq 9.8 \cdot 10^{-74}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.3e-5)
   (+ (* 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 <= 1.3e-5) {
		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 <= 1.3e-5)
		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, 1.3e-5], 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 < 1.29999999999999992e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 99.2%

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

   if 1.29999999999999992e-5 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 98.8%

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

      1. mul-1-neg 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+26} \lor \neg \left(z \leq 4.3 \cdot 10^{+101}\right) \land z \leq 3.8 \cdot 10^{+113}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 2e+26) (and (not (<= z 4.3e+101)) (<= z 3.8e+113)))
   (* x 0.5)
   (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2e+26) || (!(z <= 4.3e+101) && (z <= 3.8e+113))) {
		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 <= 2d+26) .or. (.not. (z <= 4.3d+101)) .and. (z <= 3.8d+113)) 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 <= 2e+26) || (!(z <= 4.3e+101) && (z <= 3.8e+113))) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 2e+26) or (not (z <= 4.3e+101) and (z <= 3.8e+113)):
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 2e+26) || (!(z <= 4.3e+101) && (z <= 3.8e+113)))
		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 <= 2e+26) || (~((z <= 4.3e+101)) && (z <= 3.8e+113)))
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 2e+26], And[N[Not[LessEqual[z, 4.3e+101]], $MachinePrecision], LessEqual[z, 3.8e+113]]], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.0000000000000001e26 or 4.3000000000000001e101 < z < 3.8000000000000003e113

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 49.5%

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

   if 2.0000000000000001e26 < z < 4.3000000000000001e101 or 3.8000000000000003e113 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. log-rec 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-/r* 99.9%

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

      3. fma-undefine 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. neg-mul-1 99.9%

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

      6. remove-double-neg 99.9%

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

      7. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around inf 78.8%

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

      1. mul-1-neg 78.8%

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

      2. *-commutative 78.8%

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

      3. distribute-rgt-neg-in 78.8%

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

   10. Simplified 78.8%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+26} \lor \neg \left(z \leq 4.3 \cdot 10^{+101}\right) \land z \leq 3.8 \cdot 10^{+113}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf 70.9%

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

   1. associate-*r* 70.9%

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

   2. mul-1-neg 70.9%

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

5. Simplified 70.9%

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

   1. fma-define 70.9%

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

   2. distribute-lft-neg-out 70.9%

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

   3. fmm-undef 70.9%

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

7. Applied egg-rr 70.9%

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

8. Final simplification 70.9%

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

Alternative 8: 40.0% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 38.4%

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

4. Final simplification 38.4%

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

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

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