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

Percentage Accurate: 99.9% → 99.9%

Time: 10.5s

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: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + y \cdot \log z\\ \mathbf{if}\;z \leq 2.5 \cdot 10^{-290}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \left(y \cdot \frac{1}{x}\right)\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-122}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-82}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (* y (log z)))))
   (if (<= z 2.5e-290)
     (* x (- 0.5 (* z (* y (/ 1.0 x)))))
     (if (<= z 3.2e-199)
       t_0
       (if (<= z 9.4e-122)
         (- (* x 0.5) (* y z))
         (if (<= z 2.25e-82) t_0 (fma y (- z) (* x 0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (y * log(z));
	double tmp;
	if (z <= 2.5e-290) {
		tmp = x * (0.5 - (z * (y * (1.0 / x))));
	} else if (z <= 3.2e-199) {
		tmp = t_0;
	} else if (z <= 9.4e-122) {
		tmp = (x * 0.5) - (y * z);
	} else if (z <= 2.25e-82) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(y * log(z)))
	tmp = 0.0
	if (z <= 2.5e-290)
		tmp = Float64(x * Float64(0.5 - Float64(z * Float64(y * Float64(1.0 / x)))));
	elseif (z <= 3.2e-199)
		tmp = t_0;
	elseif (z <= 9.4e-122)
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	elseif (z <= 2.25e-82)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2.5e-290], N[(x * N[(0.5 - N[(z * N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e-199], t$95$0, If[LessEqual[z, 9.4e-122], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.25e-82], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 2.5e-290

   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 80.3%

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

      1. associate-*r* 80.3%

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

      2. neg-mul-1 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in x around inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

      3. associate-/l* 80.3%

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

   8. Simplified 80.3%

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

      1. clear-num 80.3%

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

      2. un-div-inv 80.3%

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

   10. Applied egg-rr 80.3%

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

       1. associate-/r/ 80.8%

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

       2. associate-*l/ 80.3%

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

       3. clear-num 80.3%

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

       4. associate-/r/ 80.3%

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

       5. associate-*r* 80.8%

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

   12. Applied egg-rr 80.8%

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

   if 2.5e-290 < z < 3.1999999999999999e-199 or 9.3999999999999999e-122 < z < 2.2499999999999999e-82

   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-rgt-in 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 71.6%

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

   6. Taylor expanded in z around 0 71.6%

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

   if 3.1999999999999999e-199 < z < 9.3999999999999999e-122

   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 67.7%

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

      1. associate-*r* 67.7%

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

      2. neg-mul-1 67.7%

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

   5. Simplified 67.7%

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

      1. fma-define 67.7%

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

      2. distribute-lft-neg-out 67.7%

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

      3. add-sqr-sqrt 41.9%

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

      4. sqrt-unprod 63.7%

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

      5. sqr-neg 63.7%

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

      6. sqrt-unprod 25.4%

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

      7. add-sqr-sqrt 66.9%

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

      8. fma-neg 66.9%

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

      9. *-commutative 66.9%

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

      10. add-sqr-sqrt 25.4%

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

      11. sqrt-unprod 63.7%

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

      12. sqr-neg 63.7%

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

      13. sqrt-unprod 41.9%

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

      14. add-sqr-sqrt 67.7%

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

   7. Applied egg-rr 67.7%

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

   if 2.2499999999999999e-82 < 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 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. Taylor expanded in z around inf 91.1%

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

      1. neg-mul-1 91.1%

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

   7. Simplified 91.1%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-z}, x \cdot 0.5\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-290}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \left(y \cdot \frac{1}{x}\right)\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-199}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-122}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-82}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-291}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \left(y \cdot \frac{1}{x}\right)\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-199} \lor \neg \left(z \leq 1.8 \cdot 10^{-125}\right) \land z \leq 2 \cdot 10^{-82}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5e-291)
   (* x (- 0.5 (* z (* y (/ 1.0 x)))))
   (if (or (<= z 3.8e-199) (and (not (<= z 1.8e-125)) (<= z 2e-82)))
     (+ y (* y (log z)))
     (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5e-291) {
		tmp = x * (0.5 - (z * (y * (1.0 / x))));
	} else if ((z <= 3.8e-199) || (!(z <= 1.8e-125) && (z <= 2e-82))) {
		tmp = y + (y * log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 5d-291) then
        tmp = x * (0.5d0 - (z * (y * (1.0d0 / x))))
    else if ((z <= 3.8d-199) .or. (.not. (z <= 1.8d-125)) .and. (z <= 2d-82)) then
        tmp = y + (y * log(z))
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 5e-291) {
		tmp = x * (0.5 - (z * (y * (1.0 / x))));
	} else if ((z <= 3.8e-199) || (!(z <= 1.8e-125) && (z <= 2e-82))) {
		tmp = y + (y * Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5e-291:
		tmp = x * (0.5 - (z * (y * (1.0 / x))))
	elif (z <= 3.8e-199) or (not (z <= 1.8e-125) and (z <= 2e-82)):
		tmp = y + (y * math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5e-291)
		tmp = Float64(x * Float64(0.5 - Float64(z * Float64(y * Float64(1.0 / x)))));
	elseif ((z <= 3.8e-199) || (!(z <= 1.8e-125) && (z <= 2e-82)))
		tmp = Float64(y + Float64(y * log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 5e-291)
		tmp = x * (0.5 - (z * (y * (1.0 / x))));
	elseif ((z <= 3.8e-199) || (~((z <= 1.8e-125)) && (z <= 2e-82)))
		tmp = y + (y * log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5e-291], N[(x * N[(0.5 - N[(z * N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 3.8e-199], And[N[Not[LessEqual[z, 1.8e-125]], $MachinePrecision], LessEqual[z, 2e-82]]], N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.0000000000000003e-291

   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 80.3%

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

      1. associate-*r* 80.3%

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

      2. neg-mul-1 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in x around inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

      3. associate-/l* 80.3%

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

   8. Simplified 80.3%

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

      1. clear-num 80.3%

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

      2. un-div-inv 80.3%

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

   10. Applied egg-rr 80.3%

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

       1. associate-/r/ 80.8%

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

       2. associate-*l/ 80.3%

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

       3. clear-num 80.3%

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

       4. associate-/r/ 80.3%

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

       5. associate-*r* 80.8%

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

   12. Applied egg-rr 80.8%

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

   if 5.0000000000000003e-291 < z < 3.7999999999999998e-199 or 1.8000000000000001e-125 < z < 2e-82

   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-rgt-in 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 71.6%

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

   6. Taylor expanded in z around 0 71.6%

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

   if 3.7999999999999998e-199 < z < 1.8000000000000001e-125 or 2e-82 < z

   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 87.7%

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

      1. associate-*r* 87.7%

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

      2. neg-mul-1 87.7%

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

   5. Simplified 87.7%

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

      1. fma-define 87.7%

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

      2. distribute-lft-neg-out 87.7%

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

      3. add-sqr-sqrt 41.5%

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

      4. sqrt-unprod 56.0%

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

      5. sqr-neg 56.0%

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

      6. sqrt-unprod 20.9%

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

      7. add-sqr-sqrt 40.6%

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

      8. fma-neg 40.6%

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

      9. *-commutative 40.6%

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

      10. add-sqr-sqrt 20.9%

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

      11. sqrt-unprod 56.0%

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

      12. sqr-neg 56.0%

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

      13. sqrt-unprod 41.5%

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

      14. add-sqr-sqrt 87.7%

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

   7. Applied egg-rr 87.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-291}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \left(y \cdot \frac{1}{x}\right)\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-199} \lor \neg \left(z \leq 1.8 \cdot 10^{-125}\right) \land z \leq 2 \cdot 10^{-82}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49} \lor \neg \left(y \leq 6.3 \cdot 10^{-24}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.45e+49) (not (<= y 6.3e-24)))
   (* y (- (+ 1.0 (log z)) z))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.45e+49) || !(y <= 6.3e-24)) {
		tmp = y * ((1.0 + log(z)) - z);
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.45e+49) || !(y <= 6.3e-24))
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.45e+49], N[Not[LessEqual[y, 6.3e-24]], $MachinePrecision]], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e49 or 6.29999999999999979e-24 < 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 x around -inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. *-commutative 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in y around -inf 86.8%

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

   if -1.45e49 < y < 6.29999999999999979e-24

   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 90.2%

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

      1. neg-mul-1 90.2%

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

   7. Simplified 90.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+49} \lor \neg \left(y \leq 6.3 \cdot 10^{-24}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.27:\\ \;\;\;\;x \cdot 0.5 + \left(y + y \cdot \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.27) (+ (* x 0.5) (+ y (* y (log z)))) (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.27) {
		tmp = (x * 0.5) + (y + (y * log(z)));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.27)
		tmp = Float64(Float64(x * 0.5) + Float64(y + Float64(y * log(z))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.27], N[(N[(x * 0.5), $MachinePrecision] + N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.27000000000000002

   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-rgt-in 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0 97.9%

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

   if 0.27000000000000002 < 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 99.4%

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

      1. neg-mul-1 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.27:\\ \;\;\;\;x \cdot 0.5 + \left(y + y \cdot \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.27:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.27) (+ (* x 0.5) (* y (+ 1.0 (log z)))) (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.27) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.27)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.27], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.27000000000000002

   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 97.9%

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

      1. *-commutative 97.9%

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

   5. Simplified 97.9%

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

   if 0.27000000000000002 < 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 99.4%

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

      1. neg-mul-1 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.27:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 8: 59.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 5.2e+43) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.2e+43) {
		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 <= 5.2d+43) 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 <= 5.2e+43) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5.2e+43:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.2e+43)
		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 <= 5.2e+43)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.2e+43], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.20000000000000042e43

   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 56.6%

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

      1. associate-*r* 56.6%

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

      2. neg-mul-1 56.6%

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

   5. Simplified 56.6%

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

   6. Taylor expanded in x around inf 50.4%

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

   if 5.20000000000000042e43 < 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 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 72.8%

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

      1. mul-1-neg 72.8%

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

      2. *-commutative 72.8%

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

      3. distribute-rgt-neg-in 72.8%

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

   8. Simplified 72.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% 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 74.5%

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

   1. associate-*r* 74.5%

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

   2. neg-mul-1 74.5%

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

5. Simplified 74.5%

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

   1. fma-define 74.5%

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

   2. distribute-lft-neg-out 74.5%

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

   3. add-sqr-sqrt 35.7%

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

   4. sqrt-unprod 50.5%

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

   5. sqr-neg 50.5%

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

   6. sqrt-unprod 20.2%

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

   7. add-sqr-sqrt 39.7%

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

   8. fma-neg 39.7%

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

   9. *-commutative 39.7%

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

   10. add-sqr-sqrt 20.2%

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

   11. sqrt-unprod 50.5%

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

   12. sqr-neg 50.5%

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

   13. sqrt-unprod 35.7%

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

   14. add-sqr-sqrt 74.5%

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

7. Applied egg-rr 74.5%

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

8. Final simplification 74.5%

   \[\leadsto x \cdot 0.5 - y \cdot z \]
9. 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 z around inf 74.5%

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

   1. associate-*r* 74.5%

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

   2. neg-mul-1 74.5%

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

5. Simplified 74.5%

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

6. Taylor expanded in x around inf 40.8%

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

7. Final simplification 40.8%

   \[\leadsto x \cdot 0.5 \]
8. 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 2024135 
(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)))))