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

Percentage Accurate: 99.9% → 99.9%

Time: 12.5s

Alternatives: 13

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 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(x * 0.5 + N[(y * N[(1.0 + N[(N[Log[z], $MachinePrecision] - 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. Step-by-step derivation

   1. cancel-sign-sub 99.8%

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

   2. fma-neg 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. remove-double-neg 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r- 99.8%

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

   7. +-commutative 99.8%

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

   8. associate--l+ 99.8%

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

3. Simplified 99.8%

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

Alternative 3: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -5e-40)
   (* x (- 0.5 (* z (/ y x))))
   (if (<= (* x 0.5) 5e+21) (+ y (* y (- (log z) z))) (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -5e-40) {
		tmp = x * (0.5 - (z * (y / x)));
	} else if ((x * 0.5) <= 5e+21) {
		tmp = y + (y * (log(z) - 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 ((x * 0.5d0) <= (-5d-40)) then
        tmp = x * (0.5d0 - (z * (y / x)))
    else if ((x * 0.5d0) <= 5d+21) then
        tmp = y + (y * (log(z) - 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 ((x * 0.5) <= -5e-40) {
		tmp = x * (0.5 - (z * (y / x)));
	} else if ((x * 0.5) <= 5e+21) {
		tmp = y + (y * (Math.log(z) - z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * 0.5) <= -5e-40:
		tmp = x * (0.5 - (z * (y / x)))
	elif (x * 0.5) <= 5e+21:
		tmp = y + (y * (math.log(z) - z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -5e-40)
		tmp = Float64(x * Float64(0.5 - Float64(z * Float64(y / x))));
	elseif (Float64(x * 0.5) <= 5e+21)
		tmp = Float64(y + Float64(y * Float64(log(z) - 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 ((x * 0.5) <= -5e-40)
		tmp = x * (0.5 - (z * (y / x)));
	elseif ((x * 0.5) <= 5e+21)
		tmp = y + (y * (log(z) - z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -5e-40], N[(x * N[(0.5 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 5e+21], N[(y + N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -4.99999999999999965e-40

   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 79.1%

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

      1. neg-mul-1 79.1%

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

   7. Simplified 79.1%

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

   8. Taylor expanded in x around inf 79.1%

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

      1. associate-*r/ 79.1%

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

      2. mul-1-neg 79.1%

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

      3. *-commutative 79.1%

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

      4. distribute-rgt-neg-in 79.1%

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

   10. Simplified 79.1%

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

       1. frac-2neg 79.1%

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

       2. *-commutative 79.1%

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

       3. neg-mul-1 79.1%

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

       4. associate-*r* 79.1%

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

       5. distribute-frac-neg 79.1%

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

       6. associate-*r* 79.1%

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

       7. neg-mul-1 79.1%

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

       8. add-sqr-sqrt 36.8%

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

       9. sqrt-unprod 66.1%

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

       10. sqr-neg 66.1%

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

       11. sqrt-unprod 32.8%

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

       12. add-sqr-sqrt 62.3%

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

       13. remove-double-neg 62.3%

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

       14. mul-1-neg 62.3%

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

       15. frac-2neg 62.3%

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

       16. associate-*r* 62.3%

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

       17. neg-mul-1 62.3%

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

       18. *-commutative 62.3%

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

       19. associate-/l* 62.3%

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

   12. Applied egg-rr 80.3%

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

   if -4.99999999999999965e-40 < (*.f64 x #s(literal 1/2 binary64)) < 5e21

   1. Initial program 99.7%

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

      1. cancel-sign-sub 99.7%

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

      2. fma-neg 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

      4. remove-double-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      7. +-commutative 99.7%

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

      8. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. distribute-rgt-in 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 89.9%

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

   if 5e21 < (*.f64 x #s(literal 1/2 binary64))

   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 92.7%

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

      1. neg-mul-1 92.7%

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

   7. Simplified 92.7%

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

   8. Taylor expanded in y around 0 92.7%

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

      1. +-commutative 92.7%

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

      2. associate-*r* 92.7%

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

      3. mul-1-neg 92.7%

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

      4. cancel-sign-sub-inv 92.7%

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

   10. Simplified 92.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -5e-40)
   (* x (- 0.5 (* z (/ y x))))
   (if (<= (* x 0.5) 5e+21)
     (* y (+ 1.0 (- (log z) z)))
     (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -5e-40) {
		tmp = x * (0.5 - (z * (y / x)));
	} else if ((x * 0.5) <= 5e+21) {
		tmp = y * (1.0 + (log(z) - 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 ((x * 0.5d0) <= (-5d-40)) then
        tmp = x * (0.5d0 - (z * (y / x)))
    else if ((x * 0.5d0) <= 5d+21) then
        tmp = y * (1.0d0 + (log(z) - 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 ((x * 0.5) <= -5e-40) {
		tmp = x * (0.5 - (z * (y / x)));
	} else if ((x * 0.5) <= 5e+21) {
		tmp = y * (1.0 + (Math.log(z) - z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * 0.5) <= -5e-40:
		tmp = x * (0.5 - (z * (y / x)))
	elif (x * 0.5) <= 5e+21:
		tmp = y * (1.0 + (math.log(z) - z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -5e-40)
		tmp = Float64(x * Float64(0.5 - Float64(z * Float64(y / x))));
	elseif (Float64(x * 0.5) <= 5e+21)
		tmp = Float64(y * Float64(1.0 + Float64(log(z) - 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 ((x * 0.5) <= -5e-40)
		tmp = x * (0.5 - (z * (y / x)));
	elseif ((x * 0.5) <= 5e+21)
		tmp = y * (1.0 + (log(z) - z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -5e-40], N[(x * N[(0.5 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 5e+21], N[(y * N[(1.0 + N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -4.99999999999999965e-40

   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 79.1%

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

      1. neg-mul-1 79.1%

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

   7. Simplified 79.1%

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

   8. Taylor expanded in x around inf 79.1%

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

      1. associate-*r/ 79.1%

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

      2. mul-1-neg 79.1%

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

      3. *-commutative 79.1%

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

      4. distribute-rgt-neg-in 79.1%

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

   10. Simplified 79.1%

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

       1. frac-2neg 79.1%

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

       2. *-commutative 79.1%

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

       3. neg-mul-1 79.1%

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

       4. associate-*r* 79.1%

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

       5. distribute-frac-neg 79.1%

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

       6. associate-*r* 79.1%

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

       7. neg-mul-1 79.1%

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

       8. add-sqr-sqrt 36.8%

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

       9. sqrt-unprod 66.1%

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

       10. sqr-neg 66.1%

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

       11. sqrt-unprod 32.8%

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

       12. add-sqr-sqrt 62.3%

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

       13. remove-double-neg 62.3%

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

       14. mul-1-neg 62.3%

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

       15. frac-2neg 62.3%

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

       16. associate-*r* 62.3%

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

       17. neg-mul-1 62.3%

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

       18. *-commutative 62.3%

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

       19. associate-/l* 62.3%

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

   12. Applied egg-rr 80.3%

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

   if -4.99999999999999965e-40 < (*.f64 x #s(literal 1/2 binary64)) < 5e21

   1. Initial program 99.7%

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

      1. cancel-sign-sub 99.7%

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

      2. fma-neg 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

      4. remove-double-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      7. +-commutative 99.7%

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

      8. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l+ 99.7%

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

      6. +-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. +-commutative 99.7%

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

      10. *-un-lft-identity 99.7%

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

      11. distribute-rgt-in 98.9%

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

      12. associate-+l+ 98.9%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in y around inf 89.8%

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

      1. associate--l+ 89.9%

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

   9. Simplified 89.9%

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

   if 5e21 < (*.f64 x #s(literal 1/2 binary64))

   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 92.7%

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

      1. neg-mul-1 92.7%

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

   7. Simplified 92.7%

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

   8. Taylor expanded in y around 0 92.7%

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

      1. +-commutative 92.7%

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

      2. associate-*r* 92.7%

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

      3. mul-1-neg 92.7%

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

      4. cancel-sign-sub-inv 92.7%

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

   10. Simplified 92.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.4 \cdot 10^{-236}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 6.4e-236)
   (* x (- 0.5 (* z (/ y x))))
   (if (<= z 1.95e-41) (* y (+ 1.0 (log z))) (fma y (- z) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.4e-236) {
		tmp = x * (0.5 - (z * (y / x)));
	} else if (z <= 1.95e-41) {
		tmp = 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 <= 6.4e-236)
		tmp = Float64(x * Float64(0.5 - Float64(z * Float64(y / x))));
	elseif (z <= 1.95e-41)
		tmp = 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, 6.4e-236], N[(x * N[(0.5 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e-41], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 6.3999999999999999e-236

   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 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 65.6%

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

      1. neg-mul-1 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in x around inf 65.6%

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

      1. associate-*r/ 65.6%

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

      2. mul-1-neg 65.6%

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

      3. *-commutative 65.6%

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

      4. distribute-rgt-neg-in 65.6%

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

   10. Simplified 65.6%

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

       1. frac-2neg 65.6%

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

       2. *-commutative 65.6%

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

       3. neg-mul-1 65.6%

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

       4. associate-*r* 65.6%

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

       5. distribute-frac-neg 65.6%

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

       6. associate-*r* 65.6%

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

       7. neg-mul-1 65.6%

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

       8. add-sqr-sqrt 28.0%

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

       9. sqrt-unprod 65.6%

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

       10. sqr-neg 65.6%

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

       11. sqrt-unprod 37.5%

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

       12. add-sqr-sqrt 65.4%

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

       13. remove-double-neg 65.4%

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

       14. mul-1-neg 65.4%

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

       15. frac-2neg 65.4%

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

       16. associate-*r* 65.4%

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

       17. neg-mul-1 65.4%

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

       18. *-commutative 65.4%

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

       19. associate-/l* 65.4%

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

   12. Applied egg-rr 65.8%

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

   if 6.3999999999999999e-236 < z < 1.94999999999999995e-41

   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 99.7%

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

      1. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 85.4%

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

   7. Taylor expanded in x around 0 61.8%

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

   if 1.94999999999999995e-41 < 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 92.0%

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

      1. neg-mul-1 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.4 \cdot 10^{-236}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 10^{-235}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1e-235)
   (* x (- 0.5 (* z (/ y x))))
   (if (<= z 2.6e-41) (* y (+ 1.0 (log z))) (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1e-235) {
		tmp = x * (0.5 - (z * (y / x)));
	} else if (z <= 2.6e-41) {
		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 <= 1d-235) then
        tmp = x * (0.5d0 - (z * (y / x)))
    else if (z <= 2.6d-41) 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 <= 1e-235) {
		tmp = x * (0.5 - (z * (y / x)));
	} else if (z <= 2.6e-41) {
		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 <= 1e-235:
		tmp = x * (0.5 - (z * (y / x)))
	elif z <= 2.6e-41:
		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 <= 1e-235)
		tmp = Float64(x * Float64(0.5 - Float64(z * Float64(y / x))));
	elseif (z <= 2.6e-41)
		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 <= 1e-235)
		tmp = x * (0.5 - (z * (y / x)));
	elseif (z <= 2.6e-41)
		tmp = 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, 1e-235], N[(x * N[(0.5 - N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e-41], 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 3 regimes

2. if z < 9.9999999999999996e-236

   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 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 65.6%

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

      1. neg-mul-1 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in x around inf 65.6%

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

      1. associate-*r/ 65.6%

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

      2. mul-1-neg 65.6%

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

      3. *-commutative 65.6%

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

      4. distribute-rgt-neg-in 65.6%

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

   10. Simplified 65.6%

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

       1. frac-2neg 65.6%

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

       2. *-commutative 65.6%

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

       3. neg-mul-1 65.6%

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

       4. associate-*r* 65.6%

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

       5. distribute-frac-neg 65.6%

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

       6. associate-*r* 65.6%

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

       7. neg-mul-1 65.6%

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

       8. add-sqr-sqrt 28.0%

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

       9. sqrt-unprod 65.6%

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

       10. sqr-neg 65.6%

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

       11. sqrt-unprod 37.5%

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

       12. add-sqr-sqrt 65.4%

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

       13. remove-double-neg 65.4%

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

       14. mul-1-neg 65.4%

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

       15. frac-2neg 65.4%

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

       16. associate-*r* 65.4%

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

       17. neg-mul-1 65.4%

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

       18. *-commutative 65.4%

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

       19. associate-/l* 65.4%

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

   12. Applied egg-rr 65.8%

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

   if 9.9999999999999996e-236 < z < 2.5999999999999999e-41

   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 99.7%

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

      1. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 85.4%

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

   7. Taylor expanded in x around 0 61.8%

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

   if 2.5999999999999999e-41 < 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 92.0%

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

      1. neg-mul-1 92.0%

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

   7. Simplified 92.0%

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

   8. Taylor expanded in y around 0 92.0%

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

      1. +-commutative 92.0%

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

      2. associate-*r* 92.0%

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

      3. mul-1-neg 92.0%

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

      4. cancel-sign-sub-inv 92.0%

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

   10. Simplified 92.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-235}:\\ \;\;\;\;x \cdot \left(0.5 - z \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.28)
		tmp = Float64(Float64(y + Float64(x * 0.5)) + 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, 0.28], N[(N[(y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   1. Initial program 99.7%

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

      1. cancel-sign-sub 99.7%

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

      2. fma-neg 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

      4. remove-double-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r- 99.7%

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

      7. +-commutative 99.7%

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

      8. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around 0 98.8%

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

   if 0.28000000000000003 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-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 98.2%

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

      1. neg-mul-1 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 98.5%

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

Alternative 8: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   if 0.28000000000000003 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-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 98.2%

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

      1. neg-mul-1 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 98.5%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y + (x * 0.5)) + (y * (log(z) - 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 + (x * 0.5d0)) + (y * (log(z) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $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. cancel-sign-sub 99.8%

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

   2. fma-neg 99.8%

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

   3. distribute-lft-neg-in 99.8%

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

   4. remove-double-neg 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r- 99.8%

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

   7. +-commutative 99.8%

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

   8. associate--l+ 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-un-lft-identity 99.8%

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

   4. associate-+r+ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 10: 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 11: 60.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.8e+16) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.8e+16) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.8d+16) then
        tmp = x * 0.5d0
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.8e+16) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.8e+16:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.8e+16)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.8e+16)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.8e+16], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.8e16

   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. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. sub-neg 99.7%

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

      5. add-cbrt-cube 99.4%

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

      6. pow3 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around inf 47.4%

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

   if 2.8e16 < 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. cancel-sign-sub 100.0%

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

      2. fma-neg 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+r- 100.0%

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

      7. +-commutative 100.0%

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

      8. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 70.3%

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

      1. associate-*r* 70.3%

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

      2. mul-1-neg 70.3%

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

   7. Simplified 70.3%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.3% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around inf 71.9%

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

   1. neg-mul-1 71.9%

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

7. Simplified 71.9%

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

8. Taylor expanded in y around 0 71.9%

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

   1. +-commutative 71.9%

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

   2. associate-*r* 71.9%

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

   3. mul-1-neg 71.9%

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

   4. cancel-sign-sub-inv 71.9%

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

10. Simplified 71.9%

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

11. Final simplification 71.9%

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

Alternative 13: 40.3% 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. Step-by-step derivation

   1. sub-neg 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. sub-neg 99.8%

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

   5. add-cbrt-cube 79.4%

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

   6. pow3 79.4%

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

4. Applied egg-rr 79.4%

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

5. Taylor expanded in x around inf 40.6%

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

6. Final simplification 40.6%

   \[\leadsto x \cdot 0.5 \]
7. 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 2024123 
(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)))))