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

Percentage Accurate: 99.9% → 99.9%

Time: 35.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+41} \lor \neg \left(y \leq 8.2 \cdot 10^{+52}\right):\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.3e+41) (not (<= y 8.2e+52)))
   (* y (+ (- 1.0 z) (log z)))
   (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e+41) || !(y <= 8.2e+52)) {
		tmp = y * ((1.0 - z) + 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 ((y <= (-1.3d+41)) .or. (.not. (y <= 8.2d+52))) then
        tmp = y * ((1.0d0 - z) + 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 ((y <= -1.3e+41) || !(y <= 8.2e+52)) {
		tmp = y * ((1.0 - z) + Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.3e+41) or not (y <= 8.2e+52):
		tmp = y * ((1.0 - z) + math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.3e+41) || !(y <= 8.2e+52))
		tmp = Float64(y * Float64(Float64(1.0 - z) + 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 ((y <= -1.3e+41) || ~((y <= 8.2e+52)))
		tmp = y * ((1.0 - z) + log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.3e+41], N[Not[LessEqual[y, 8.2e+52]], $MachinePrecision]], N[(y * N[(N[(1.0 - z), $MachinePrecision] + 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 y < -1.3e41 or 8.1999999999999999e52 < y

   1. Initial program 99.7%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   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 inf 69.5%

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

   6. Taylor expanded in x around 0 87.1%

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

      1. distribute-lft-out 87.1%

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

   8. Simplified 87.1%

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

   if -1.3e41 < y < 8.1999999999999999e52

   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 89.2%

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

      1. associate-*r* 89.2%

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

      2. mul-1-neg 89.2%

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

   5. Simplified 89.2%

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

      1. fma-define 89.2%

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

      2. distribute-lft-neg-out 89.2%

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

      3. add-sqr-sqrt 42.4%

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

      4. sqrt-unprod 74.7%

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

      5. sqr-neg 74.7%

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

      6. sqrt-unprod 33.6%

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

      7. add-sqr-sqrt 66.4%

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

      8. fma-neg 66.4%

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

      9. add-sqr-sqrt 33.6%

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

      10. sqrt-unprod 74.7%

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

      11. sqr-neg 74.7%

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

      12. sqrt-unprod 42.4%

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

      13. add-sqr-sqrt 89.2%

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

   7. Applied egg-rr 89.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+41} \lor \neg \left(y \leq 8.2 \cdot 10^{+52}\right):\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+41} \lor \neg \left(y \leq 1.46 \cdot 10^{+162}\right):\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7e+41) (not (<= y 1.46e+162)))
   (* y (+ 1.0 (log z)))
   (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7e+41) || !(y <= 1.46e+162)) {
		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 ((y <= (-7d+41)) .or. (.not. (y <= 1.46d+162))) 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 ((y <= -7e+41) || !(y <= 1.46e+162)) {
		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 (y <= -7e+41) or not (y <= 1.46e+162):
		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 ((y <= -7e+41) || !(y <= 1.46e+162))
		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 ((y <= -7e+41) || ~((y <= 1.46e+162)))
		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[y, -7e+41], N[Not[LessEqual[y, 1.46e+162]], $MachinePrecision]], 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 y < -6.9999999999999998e41 or 1.4599999999999999e162 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 67.9%

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

   4. Taylor expanded in x around 0 57.8%

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

   if -6.9999999999999998e41 < y < 1.4599999999999999e162

   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 86.7%

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

      1. associate-*r* 86.7%

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

      2. mul-1-neg 86.7%

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

   5. Simplified 86.7%

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

      1. fma-define 86.7%

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

      2. distribute-lft-neg-out 86.7%

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

      3. add-sqr-sqrt 47.2%

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

      4. sqrt-unprod 73.9%

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

      5. sqr-neg 73.9%

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

      6. sqrt-unprod 28.4%

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

      7. add-sqr-sqrt 58.9%

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

      8. fma-neg 58.9%

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

      9. add-sqr-sqrt 28.4%

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

      10. sqrt-unprod 73.9%

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

      11. sqr-neg 73.9%

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

      12. sqrt-unprod 47.2%

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

      13. add-sqr-sqrt 86.7%

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

   7. Applied egg-rr 86.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+41} \lor \neg \left(y \leq 1.46 \cdot 10^{+162}\right):\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+162}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.15e+42)
   (* y (+ 1.0 (log z)))
   (if (<= y 1.32e+162) (- (* x 0.5) (* y z)) (+ y (* y (log z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e+42) {
		tmp = y * (1.0 + log(z));
	} else if (y <= 1.32e+162) {
		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 (y <= (-1.15d+42)) then
        tmp = y * (1.0d0 + log(z))
    else if (y <= 1.32d+162) 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 (y <= -1.15e+42) {
		tmp = y * (1.0 + Math.log(z));
	} else if (y <= 1.32e+162) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y + (y * Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.15e+42:
		tmp = y * (1.0 + math.log(z))
	elif y <= 1.32e+162:
		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 (y <= -1.15e+42)
		tmp = Float64(y * Float64(1.0 + log(z)));
	elseif (y <= 1.32e+162)
		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 (y <= -1.15e+42)
		tmp = y * (1.0 + log(z));
	elseif (y <= 1.32e+162)
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y + (y * log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.15e+42], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.32e+162], 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 3 regimes

2. if y < -1.15e42

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 73.0%

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

   4. Taylor expanded in x around 0 60.3%

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

   if -1.15e42 < y < 1.31999999999999999e162

   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 86.7%

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

      1. associate-*r* 86.7%

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

      2. mul-1-neg 86.7%

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

   5. Simplified 86.7%

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

      1. fma-define 86.7%

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

      2. distribute-lft-neg-out 86.7%

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

      3. add-sqr-sqrt 47.2%

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

      4. sqrt-unprod 73.9%

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

      5. sqr-neg 73.9%

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

      6. sqrt-unprod 28.4%

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

      7. add-sqr-sqrt 58.9%

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

      8. fma-neg 58.9%

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

      9. add-sqr-sqrt 28.4%

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

      10. sqrt-unprod 73.9%

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

      11. sqr-neg 73.9%

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

      12. sqrt-unprod 47.2%

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

      13. add-sqr-sqrt 86.7%

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

   7. Applied egg-rr 86.7%

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

   if 1.31999999999999999e162 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 61.6%

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

   4. Taylor expanded in x around 0 54.9%

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

      1. +-commutative 54.9%

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

      2. distribute-lft-in 55.0%

         \[\leadsto \color{blue}{y \cdot \log z + y \cdot 1} \]

      3. *-rgt-identity 55.0%

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

   6. Applied egg-rr 55.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+162}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0031:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.0031) (+ (* x 0.5) (* y (+ 1.0 (log z)))) (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.0031) {
		tmp = (x * 0.5) + (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 <= 0.0031d0) then
        tmp = (x * 0.5d0) + (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 <= 0.0031) {
		tmp = (x * 0.5) + (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 <= 0.0031:
		tmp = (x * 0.5) + (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 <= 0.0031)
		tmp = Float64(Float64(x * 0.5) + 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 <= 0.0031)
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.0031], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + 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 < 0.00309999999999999989

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 99.0%

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

   if 0.00309999999999999989 < 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 98.8%

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

      1. associate-*r* 98.8%

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

      2. mul-1-neg 98.8%

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

   5. Simplified 98.8%

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

      1. fma-define 98.8%

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

      2. distribute-lft-neg-out 98.8%

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

      3. add-sqr-sqrt 54.5%

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

      4. sqrt-unprod 67.7%

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

      5. sqr-neg 67.7%

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

      6. sqrt-unprod 16.0%

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

      7. add-sqr-sqrt 33.1%

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

      8. fma-neg 33.1%

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

      9. add-sqr-sqrt 16.0%

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

      10. sqrt-unprod 67.7%

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

      11. sqr-neg 67.7%

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

      12. sqrt-unprod 54.5%

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

      13. add-sqr-sqrt 98.8%

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

   7. Applied egg-rr 98.8%

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

Alternative 6: 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. Add Preprocessing
3. Add Preprocessing

Alternative 7: 57.4% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+111}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.8e+111) (* x 0.5) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e+111) {
		tmp = x * 0.5;
	} else {
		tmp = y * (1.0 - 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 <= 1.8d+111) then
        tmp = x * 0.5d0
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e+111) {
		tmp = x * 0.5;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.8e+111:
		tmp = x * 0.5
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.8e+111)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.8e+111)
		tmp = x * 0.5;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.8e+111], N[(x * 0.5), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.8000000000000001e111

   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 52.4%

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

   if 1.8000000000000001e111 < 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 x around -inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. distribute-rgt-neg-in 90.0%

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

      3. sub-neg 90.0%

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

      4. mul-1-neg 90.0%

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

      5. *-commutative 90.0%

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

      6. +-commutative 90.0%

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

      7. associate--l+ 90.0%

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

      8. +-commutative 90.0%

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

      9. associate-/l* 89.8%

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

      10. distribute-lft-neg-in 89.8%

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

      11. associate-+l- 89.8%

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

      12. sub-neg 89.8%

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

      13. distribute-neg-in 89.8%

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

      14. metadata-eval 89.8%

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

      15. remove-double-neg 89.8%

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

   5. Simplified 89.8%

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

   6. Taylor expanded in z around inf 89.8%

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

   7. Taylor expanded in x around 0 82.9%

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

      1. mul-1-neg 82.9%

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

      2. sub-neg 82.9%

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

      3. metadata-eval 82.9%

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

      4. distribute-rgt-neg-in 82.9%

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

      5. +-commutative 82.9%

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

      6. distribute-neg-in 82.9%

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

      7. metadata-eval 82.9%

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

      8. sub-neg 82.9%

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

   9. Simplified 82.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+111}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.4% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+111}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.8e+111) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e+111) {
		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 <= 1.8d+111) 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 <= 1.8e+111) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.8e+111:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.8e+111)
		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 <= 1.8e+111)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.8e+111], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.8000000000000001e111

   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 52.4%

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

   if 1.8000000000000001e111 < z

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 79.4%

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

   6. Taylor expanded in z around inf 82.9%

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

      1. neg-mul-1 82.9%

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

      2. distribute-rgt-neg-in 82.9%

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

   8. Simplified 82.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+111}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.5% 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 99.8%

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

3. Taylor expanded in z around inf 71.9%

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

   1. associate-*r* 71.9%

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

   2. mul-1-neg 71.9%

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

5. Simplified 71.9%

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

   1. fma-define 71.9%

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

   2. distribute-lft-neg-out 71.9%

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

   3. add-sqr-sqrt 37.8%

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

   4. sqrt-unprod 57.3%

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

   5. sqr-neg 57.3%

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

   6. sqrt-unprod 21.6%

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

   7. add-sqr-sqrt 42.9%

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

   8. fma-neg 42.9%

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

   9. add-sqr-sqrt 21.6%

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

   10. sqrt-unprod 57.3%

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

   11. sqr-neg 57.3%

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

   12. sqrt-unprod 37.8%

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

   13. add-sqr-sqrt 71.9%

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

7. Applied egg-rr 71.9%

   \[\leadsto \color{blue}{x \cdot 0.5 - y \cdot z} \]
8. Add Preprocessing

Alternative 10: 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 99.8%

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

3. Taylor expanded in x around inf 43.8%

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

4. Final simplification 43.8%

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

Developer Target 1: 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 2024116 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ y (* 1/2 x)) (* y (- z (log z)))))

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