Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.5s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.26 \cdot 10^{+62}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+130} \lor \neg \left(y \leq 2.2 \cdot 10^{+188}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.26e+62)
   (- (+ x (* (log y) -0.5)) z)
   (if (or (<= y 8.2e+130) (not (<= y 2.2e+188)))
     (- (* y (- 1.0 (log y))) z)
     (+ y (- x (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.26e+62) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if ((y <= 8.2e+130) || !(y <= 2.2e+188)) {
		tmp = (y * (1.0 - log(y))) - z;
	} else {
		tmp = y + (x - (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.26d+62) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if ((y <= 8.2d+130) .or. (.not. (y <= 2.2d+188))) then
        tmp = (y * (1.0d0 - log(y))) - z
    else
        tmp = y + (x - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.26e+62) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if ((y <= 8.2e+130) || !(y <= 2.2e+188)) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else {
		tmp = y + (x - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.26e+62:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif (y <= 8.2e+130) or not (y <= 2.2e+188):
		tmp = (y * (1.0 - math.log(y))) - z
	else:
		tmp = y + (x - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.26e+62)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif ((y <= 8.2e+130) || !(y <= 2.2e+188))
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	else
		tmp = Float64(y + Float64(x - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.26e+62)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif ((y <= 8.2e+130) || ~((y <= 2.2e+188)))
		tmp = (y * (1.0 - log(y))) - z;
	else
		tmp = y + (x - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.26e+62], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 8.2e+130], N[Not[LessEqual[y, 2.2e+188]], $MachinePrecision]], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.25999999999999995e62

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-def 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 96.2%

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

      1. +-commutative 96.2%

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

   6. Simplified 96.2%

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

   if 1.25999999999999995e62 < y < 8.19999999999999955e130 or 2.19999999999999999e188 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 89.6%

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

      1. *-commutative 89.6%

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

      2. log-rec 89.6%

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

      3. distribute-lft-neg-in 89.6%

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

      4. distribute-rgt-neg-in 89.6%

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

   6. Simplified 89.6%

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

   7. Taylor expanded in y around 0 89.7%

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

      1. +-commutative 89.7%

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

      2. mul-1-neg 89.7%

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

      3. unsub-neg 89.7%

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

      4. neg-mul-1 89.7%

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

      5. sub-neg 89.7%

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

   9. Simplified 89.7%

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

   if 8.19999999999999955e130 < y < 2.19999999999999999e188

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. associate-+r+ 99.7%

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

      2. associate-*r* 99.7%

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

      3. neg-mul-1 99.7%

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

      4. +-commutative 99.7%

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

      5. cancel-sign-sub-inv 99.7%

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

      6. associate--r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. associate--l+ 99.7%

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

      11. +-commutative 99.7%

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

      12. +-commutative 99.7%

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

      13. fma-def 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. log-rec 83.7%

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

      3. distribute-rgt-neg-in 83.7%

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

      4. remove-double-neg 83.7%

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

   9. Simplified 83.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.26 \cdot 10^{+62}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+130} \lor \neg \left(y \leq 2.2 \cdot 10^{+188}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \end{array} \]

Alternative 3: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-228}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-212}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;y + \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e-228)
   (- x z)
   (if (<= y 2.25e-212)
     (- x (* (log y) 0.5))
     (if (<= y 6.5e+15) (+ y (- x z)) (+ y (- x (* y (log y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-228) {
		tmp = x - z;
	} else if (y <= 2.25e-212) {
		tmp = x - (log(y) * 0.5);
	} else if (y <= 6.5e+15) {
		tmp = y + (x - z);
	} else {
		tmp = y + (x - (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.2d-228) then
        tmp = x - z
    else if (y <= 2.25d-212) then
        tmp = x - (log(y) * 0.5d0)
    else if (y <= 6.5d+15) then
        tmp = y + (x - z)
    else
        tmp = y + (x - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-228) {
		tmp = x - z;
	} else if (y <= 2.25e-212) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (y <= 6.5e+15) {
		tmp = y + (x - z);
	} else {
		tmp = y + (x - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.2e-228:
		tmp = x - z
	elif y <= 2.25e-212:
		tmp = x - (math.log(y) * 0.5)
	elif y <= 6.5e+15:
		tmp = y + (x - z)
	else:
		tmp = y + (x - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e-228)
		tmp = Float64(x - z);
	elseif (y <= 2.25e-212)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (y <= 6.5e+15)
		tmp = Float64(y + Float64(x - z));
	else
		tmp = Float64(y + Float64(x - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.2e-228)
		tmp = x - z;
	elseif (y <= 2.25e-212)
		tmp = x - (log(y) * 0.5);
	elseif (y <= 6.5e+15)
		tmp = y + (x - z);
	else
		tmp = y + (x - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.2e-228], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.25e-212], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+15], N[(y + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.2e-228

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-def 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 76.0%

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

      1. log-rec 76.0%

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

      2. sub-neg 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in y around 0 76.0%

      \[\leadsto \color{blue}{x - z} \]

   if 5.2e-228 < y < 2.2499999999999999e-212

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

   if 2.2499999999999999e-212 < y < 6.5e15

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-def 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. associate-*r* 100.0%

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

      3. neg-mul-1 100.0%

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

      4. +-commutative 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. associate--r+ 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. associate--l+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. fma-def 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in z around inf 75.6%

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

   if 6.5e15 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. associate-*r* 99.6%

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

      3. neg-mul-1 99.6%

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

      4. +-commutative 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. associate--r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. associate--l+ 99.6%

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

      11. +-commutative 99.6%

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

      12. +-commutative 99.6%

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

      13. fma-def 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 77.7%

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

      1. mul-1-neg 77.7%

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

      2. log-rec 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. remove-double-neg 77.7%

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

   9. Simplified 77.7%

      \[\leadsto y + \left(x - \color{blue}{y \cdot \log y}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-228}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-212}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;y + \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \end{array} \]

Alternative 4: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-228}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-213}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+199}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.5e-228)
   (- x z)
   (if (<= y 6.6e-213)
     (- x (* (log y) 0.5))
     (if (<= y 4.2e+199) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e-228) {
		tmp = x - z;
	} else if (y <= 6.6e-213) {
		tmp = x - (log(y) * 0.5);
	} else if (y <= 4.2e+199) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.5d-228) then
        tmp = x - z
    else if (y <= 6.6d-213) then
        tmp = x - (log(y) * 0.5d0)
    else if (y <= 4.2d+199) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e-228) {
		tmp = x - z;
	} else if (y <= 6.6e-213) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (y <= 4.2e+199) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.5e-228:
		tmp = x - z
	elif y <= 6.6e-213:
		tmp = x - (math.log(y) * 0.5)
	elif y <= 4.2e+199:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.5e-228)
		tmp = Float64(x - z);
	elseif (y <= 6.6e-213)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (y <= 4.2e+199)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.5e-228)
		tmp = x - z;
	elseif (y <= 6.6e-213)
		tmp = x - (log(y) * 0.5);
	elseif (y <= 4.2e+199)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.5e-228], N[(x - z), $MachinePrecision], If[LessEqual[y, 6.6e-213], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+199], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.49999999999999952e-228 or 6.60000000000000062e-213 < y < 4.1999999999999999e199

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 85.7%

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

      1. log-rec 85.7%

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

      2. sub-neg 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in y around 0 68.3%

      \[\leadsto \color{blue}{x - z} \]

   if 5.49999999999999952e-228 < y < 6.60000000000000062e-213

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

   if 4.1999999999999999e199 < y

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. associate-+l+ 99.4%

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

      4. associate-+r- 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-def 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. associate-+r+ 99.4%

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

      2. associate-*r* 99.4%

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

      3. neg-mul-1 99.4%

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

      4. +-commutative 99.4%

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

      5. cancel-sign-sub-inv 99.4%

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

      6. associate--r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. +-commutative 99.4%

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

      9. +-commutative 99.4%

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

      10. associate--l+ 99.4%

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

      11. +-commutative 99.4%

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

      12. +-commutative 99.4%

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

      13. fma-def 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around inf 78.7%

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

      1. log-rec 78.7%

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

      2. cancel-sign-sub-inv 78.7%

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

      3. metadata-eval 78.7%

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

      4. *-lft-identity 78.7%

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

      5. log-rec 78.7%

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

      6. log-rec 78.7%

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

      7. sub-neg 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-228}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-213}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+199}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 5: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+54}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+94}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.08e+54)
   (- x z)
   (if (<= x 1.9e+94) (- (* y (- 1.0 (log y))) z) (+ y (- x (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.08e+54) {
		tmp = x - z;
	} else if (x <= 1.9e+94) {
		tmp = (y * (1.0 - log(y))) - z;
	} else {
		tmp = y + (x - (y * log(y)));
	}
	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 <= (-1.08d+54)) then
        tmp = x - z
    else if (x <= 1.9d+94) then
        tmp = (y * (1.0d0 - log(y))) - z
    else
        tmp = y + (x - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.08e+54) {
		tmp = x - z;
	} else if (x <= 1.9e+94) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else {
		tmp = y + (x - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.08e+54:
		tmp = x - z
	elif x <= 1.9e+94:
		tmp = (y * (1.0 - math.log(y))) - z
	else:
		tmp = y + (x - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.08e+54)
		tmp = Float64(x - z);
	elseif (x <= 1.9e+94)
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	else
		tmp = Float64(y + Float64(x - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.08e+54)
		tmp = x - z;
	elseif (x <= 1.9e+94)
		tmp = (y * (1.0 - log(y))) - z;
	else
		tmp = y + (x - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.08e+54], N[(x - z), $MachinePrecision], If[LessEqual[x, 1.9e+94], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08000000000000008e54

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 99.9%

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

      1. log-rec 99.9%

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

      2. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around 0 86.3%

      \[\leadsto \color{blue}{x - z} \]

   if -1.08000000000000008e54 < x < 1.8999999999999998e94

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. log-rec 75.0%

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

      3. distribute-lft-neg-in 75.0%

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

      4. distribute-rgt-neg-in 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in y around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. neg-mul-1 75.1%

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

      5. sub-neg 75.1%

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

   9. Simplified 75.1%

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

   if 1.8999999999999998e94 < x

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. associate-+r+ 99.9%

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

      2. associate-*r* 99.9%

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

      3. neg-mul-1 99.9%

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

      4. +-commutative 99.9%

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

      5. cancel-sign-sub-inv 99.9%

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

      6. associate--r+ 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. associate--l+ 99.9%

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

      11. +-commutative 99.9%

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

      12. +-commutative 99.9%

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

      13. fma-def 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around inf 95.4%

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

      1. mul-1-neg 95.4%

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

      2. log-rec 95.4%

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

      3. distribute-rgt-neg-in 95.4%

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

      4. remove-double-neg 95.4%

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

   9. Simplified 95.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+54}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+94}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \left(x - y \cdot \log y\right)\\ \end{array} \]

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e-18)
   (- (+ x (* (log y) -0.5)) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e-18) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - log(y))) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.35d-18) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e-18) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.35e-18:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e-18)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.35e-18)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e-18], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.34999999999999994e-18

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-def 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   if 1.34999999999999994e-18 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 99.4%

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

      1. log-rec 99.4%

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

      2. sub-neg 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 8: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+199}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e+199) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+199) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.5d+199) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+199) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.5e+199:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e+199)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e+199)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.5e+199], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.49999999999999981e199

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 84.1%

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

      1. log-rec 84.1%

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

      2. sub-neg 84.1%

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

   6. Simplified 84.1%

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

   7. Taylor expanded in y around 0 67.4%

      \[\leadsto \color{blue}{x - z} \]

   if 3.49999999999999981e199 < y

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. associate-+l+ 99.4%

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

      4. associate-+r- 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-def 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. associate-+r+ 99.4%

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

      2. associate-*r* 99.4%

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

      3. neg-mul-1 99.4%

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

      4. +-commutative 99.4%

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

      5. cancel-sign-sub-inv 99.4%

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

      6. associate--r+ 99.4%

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

      7. +-commutative 99.4%

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

      8. +-commutative 99.4%

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

      9. +-commutative 99.4%

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

      10. associate--l+ 99.4%

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

      11. +-commutative 99.4%

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

      12. +-commutative 99.4%

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

      13. fma-def 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around inf 78.7%

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

      1. log-rec 78.7%

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

      2. cancel-sign-sub-inv 78.7%

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

      3. metadata-eval 78.7%

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

      4. *-lft-identity 78.7%

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

      5. log-rec 78.7%

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

      6. log-rec 78.7%

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

      7. sub-neg 78.7%

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

   9. Simplified 78.7%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+199}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 9: 48.3% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+94}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e+72) x (if (<= x 6.5e+94) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e+72) {
		tmp = x;
	} else if (x <= 6.5e+94) {
		tmp = -z;
	} else {
		tmp = x;
	}
	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 <= (-1.4d+72)) then
        tmp = x
    else if (x <= 6.5d+94) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e+72) {
		tmp = x;
	} else if (x <= 6.5e+94) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4e+72:
		tmp = x
	elif x <= 6.5e+94:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e+72)
		tmp = x;
	elseif (x <= 6.5e+94)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e+72)
		tmp = x;
	elseif (x <= 6.5e+94)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e+72], x, If[LessEqual[x, 6.5e+94], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e72 or 6.49999999999999976e94 < x

   1. Initial program 99.8%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 71.2%

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

   if -1.4e72 < x < 6.49999999999999976e94

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. associate-+r+ 99.8%

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

      2. associate-*r* 99.8%

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

      3. neg-mul-1 99.8%

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

      4. +-commutative 99.8%

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

      5. cancel-sign-sub-inv 99.8%

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

      6. associate--r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

      9. +-commutative 99.8%

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

      10. associate--l+ 99.8%

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

      11. +-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. fma-def 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in z around inf 39.1%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   8. Step-by-step derivation

      1. neg-mul-1 39.1%

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

   9. Simplified 39.1%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+94}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 57.8% accurate, 37.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x - 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 - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around inf 87.2%

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

   1. log-rec 87.2%

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

   2. sub-neg 87.2%

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

6. Simplified 87.2%

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

7. Taylor expanded in y around 0 58.0%

   \[\leadsto \color{blue}{x - z} \]

8. Final simplification 58.0%

   \[\leadsto x - z \]

Alternative 11: 30.1% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 29.1%

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

5. Final simplification 29.1%

   \[\leadsto x \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) - z) - ((y + 0.5d0) * log(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023334 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

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