Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.0s

Alternatives: 10

Speedup: 1.0×

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. sub-neg 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+l+ 99.8%

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

   5. sub-neg 99.8%

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

   6. neg-sub0 99.8%

      \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

   7. associate-+l- 99.8%

      \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

   8. neg-sub0 99.8%

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

   9. neg-mul-1 99.8%

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

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: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-261}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+104} \lor \neg \left(y \leq 1.55 \cdot 10^{+134}\right) \land y \leq 2.6 \cdot 10^{+147}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e-274)
   (- x z)
   (if (<= y 2.9e-261)
     (* (log y) -0.5)
     (if (or (<= y 2.4e+104) (and (not (<= y 1.55e+134)) (<= y 2.6e+147)))
       (- x z)
       (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e-274) {
		tmp = x - z;
	} else if (y <= 2.9e-261) {
		tmp = log(y) * -0.5;
	} else if ((y <= 2.4e+104) || (!(y <= 1.55e+134) && (y <= 2.6e+147))) {
		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 <= 4.4d-274) then
        tmp = x - z
    else if (y <= 2.9d-261) then
        tmp = log(y) * (-0.5d0)
    else if ((y <= 2.4d+104) .or. (.not. (y <= 1.55d+134)) .and. (y <= 2.6d+147)) 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 <= 4.4e-274) {
		tmp = x - z;
	} else if (y <= 2.9e-261) {
		tmp = Math.log(y) * -0.5;
	} else if ((y <= 2.4e+104) || (!(y <= 1.55e+134) && (y <= 2.6e+147))) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.4e-274:
		tmp = x - z
	elif y <= 2.9e-261:
		tmp = math.log(y) * -0.5
	elif (y <= 2.4e+104) or (not (y <= 1.55e+134) and (y <= 2.6e+147)):
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e-274)
		tmp = Float64(x - z);
	elseif (y <= 2.9e-261)
		tmp = Float64(log(y) * -0.5);
	elseif ((y <= 2.4e+104) || (!(y <= 1.55e+134) && (y <= 2.6e+147)))
		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 <= 4.4e-274)
		tmp = x - z;
	elseif (y <= 2.9e-261)
		tmp = log(y) * -0.5;
	elseif ((y <= 2.4e+104) || (~((y <= 1.55e+134)) && (y <= 2.6e+147)))
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e-274], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.9e-261], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], If[Or[LessEqual[y, 2.4e+104], And[N[Not[LessEqual[y, 1.55e+134]], $MachinePrecision], LessEqual[y, 2.6e+147]]], 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 < 4.3999999999999999e-274 or 2.89999999999999985e-261 < y < 2.4e104 or 1.54999999999999991e134 < y < 2.5999999999999999e147

   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. *-commutative 99.9%

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

      2. flip-+ 99.9%

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

      3. associate-*r/ 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}{y - 0.5}\right) + y\right) - z \]

      6. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y - 0.5}\right) + y\right) - z \]

      7. sub-neg 99.9%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      8. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   3. Applied egg-rr 99.9%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}\right) + y\right) - z \]
   4. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) + y\right) - z \]

      2. associate-/r/ 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)}\right) + y\right) - z \]

      3. +-commutative 99.9%

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{-0.5 + y}} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   5. Simplified 99.9%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{-0.5 + y} \cdot \mathsf{fma}\left(y, y, -0.25\right)}\right) + y\right) - z \]
   6. Step-by-step derivation

      1. div-inv 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\left(\log y \cdot \frac{1}{-0.5 + y}\right)} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

      2. +-commutative 99.9%

         \[\leadsto \left(\left(x - \left(\log y \cdot \frac{1}{\color{blue}{y + -0.5}}\right) \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   7. Applied egg-rr 99.9%

      \[\leadsto \left(\left(x - \color{blue}{\left(\log y \cdot \frac{1}{y + -0.5}\right)} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   8. Taylor expanded in x around inf 76.3%

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

   if 4.3999999999999999e-274 < y < 2.89999999999999985e-261

   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. sub-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 100.0%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 100.0%

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

      9. neg-mul-1 100.0%

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

   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 z around 0 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. sub-neg 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in x around 0 84.9%

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

   8. Taylor expanded in y around 0 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} \]
   9. Step-by-step derivation

      1. *-commutative 84.9%

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

   10. Simplified 84.9%

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

   if 2.4e104 < y < 1.54999999999999991e134 or 2.5999999999999999e147 < 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. sub-neg 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+l+ 99.6%

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

      5. sub-neg 99.6%

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

      6. neg-sub0 99.6%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 99.6%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 99.6%

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

      9. neg-mul-1 99.6%

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

   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 z around 0 93.7%

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

      1. +-commutative 93.7%

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

      2. mul-1-neg 93.7%

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

      3. +-commutative 93.7%

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

      4. *-commutative 93.7%

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

      5. sub-neg 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in x around 0 82.2%

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

   8. Taylor expanded in y around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. log-rec 82.2%

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

      3. distribute-rgt-neg-in 82.2%

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

      4. remove-double-neg 82.2%

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

   10. Simplified 82.2%

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

   11. Taylor expanded in y around 0 82.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-261}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+104} \lor \neg \left(y \leq 1.55 \cdot 10^{+134}\right) \land y \leq 2.6 \cdot 10^{+147}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-274}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-261}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+49}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.2e-274)
   (- x z)
   (if (<= y 2.9e-261)
     (* (log y) -0.5)
     (if (<= y 3.3e+49) (- x z) (- x (* y (+ (log y) -1.0)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e-274) {
		tmp = x - z;
	} else if (y <= 2.9e-261) {
		tmp = log(y) * -0.5;
	} else if (y <= 3.3e+49) {
		tmp = x - z;
	} else {
		tmp = x - (y * (log(y) + -1.0));
	}
	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 <= 6.2d-274) then
        tmp = x - z
    else if (y <= 2.9d-261) then
        tmp = log(y) * (-0.5d0)
    else if (y <= 3.3d+49) then
        tmp = x - z
    else
        tmp = x - (y * (log(y) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e-274) {
		tmp = x - z;
	} else if (y <= 2.9e-261) {
		tmp = Math.log(y) * -0.5;
	} else if (y <= 3.3e+49) {
		tmp = x - z;
	} else {
		tmp = x - (y * (Math.log(y) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.2e-274:
		tmp = x - z
	elif y <= 2.9e-261:
		tmp = math.log(y) * -0.5
	elif y <= 3.3e+49:
		tmp = x - z
	else:
		tmp = x - (y * (math.log(y) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e-274)
		tmp = Float64(x - z);
	elseif (y <= 2.9e-261)
		tmp = Float64(log(y) * -0.5);
	elseif (y <= 3.3e+49)
		tmp = Float64(x - z);
	else
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.2e-274)
		tmp = x - z;
	elseif (y <= 2.9e-261)
		tmp = log(y) * -0.5;
	elseif (y <= 3.3e+49)
		tmp = x - z;
	else
		tmp = x - (y * (log(y) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.2e-274], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.9e-261], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[y, 3.3e+49], N[(x - z), $MachinePrecision], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.19999999999999956e-274 or 2.89999999999999985e-261 < y < 3.2999999999999998e49

   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. *-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}{y - 0.5}\right) + y\right) - z \]

      6. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y - 0.5}\right) + y\right) - z \]

      7. sub-neg 100.0%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      8. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   3. Applied egg-rr 100.0%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}\right) + y\right) - z \]
   4. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) + y\right) - z \]

      2. associate-/r/ 100.0%

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)}\right) + y\right) - z \]

      3. +-commutative 100.0%

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{-0.5 + y}} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   5. Simplified 100.0%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{-0.5 + y} \cdot \mathsf{fma}\left(y, y, -0.25\right)}\right) + y\right) - z \]
   6. Step-by-step derivation

      1. div-inv 100.0%

         \[\leadsto \left(\left(x - \color{blue}{\left(\log y \cdot \frac{1}{-0.5 + y}\right)} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

      2. +-commutative 100.0%

         \[\leadsto \left(\left(x - \left(\log y \cdot \frac{1}{\color{blue}{y + -0.5}}\right) \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   7. Applied egg-rr 100.0%

      \[\leadsto \left(\left(x - \color{blue}{\left(\log y \cdot \frac{1}{y + -0.5}\right)} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   8. Taylor expanded in x around inf 79.3%

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

   if 6.19999999999999956e-274 < y < 2.89999999999999985e-261

   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. sub-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 100.0%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 100.0%

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

      9. neg-mul-1 100.0%

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

   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 z around 0 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. sub-neg 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in x around 0 84.9%

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

   8. Taylor expanded in y around 0 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} \]
   9. Step-by-step derivation

      1. *-commutative 84.9%

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

   10. Simplified 84.9%

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

   if 3.2999999999999998e49 < y

   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. sub-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 99.7%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   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 z around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. mul-1-neg 88.6%

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

      3. +-commutative 88.6%

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

      4. *-commutative 88.6%

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

      5. sub-neg 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in y around 0 88.7%

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

      1. *-commutative 88.7%

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

      2. cancel-sign-sub-inv 88.7%

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

      3. +-commutative 88.7%

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

      4. *-commutative 88.7%

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

      5. distribute-lft-out-- 88.6%

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

      6. *-rgt-identity 88.6%

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

      7. sub-neg 88.6%

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

      8. neg-mul-1 88.6%

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

      9. associate-+r+ 88.6%

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

      10. log-rec 88.6%

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

      11. *-commutative 88.6%

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

      12. associate-+r+ 88.6%

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

      13. neg-mul-1 88.6%

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

      14. distribute-rgt-neg-in 88.6%

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

      15. log-rec 88.6%

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

      16. distribute-rgt-in 88.6%

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

      17. log-rec 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in y around inf 88.7%

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

       1. sub-neg 88.7%

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

       2. mul-1-neg 88.7%

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

       3. log-rec 88.7%

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

       4. remove-double-neg 88.7%

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

       5. metadata-eval 88.7%

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

   12. Simplified 88.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-274}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-261}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+49}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.8e-5)
   (- (- 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 <= 5.8e-5) {
		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 <= 5.8d-5) 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 <= 5.8e-5) {
		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 <= 5.8e-5:
		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 <= 5.8e-5)
		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 <= 5.8e-5)
		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, 5.8e-5], 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 < 5.8e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 5.8e-5 < y

   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. sub-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 99.7%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   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 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 6: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+50}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+50) (- (- x (* (log y) 0.5)) z) (- x (* y (+ (log y) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+50) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x - (y * (log(y) + -1.0));
	}
	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 <= 6.5d+50) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x - (y * (log(y) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+50) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x - (y * (Math.log(y) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.5e+50:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x - (y * (math.log(y) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+50)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.5e+50)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x - (y * (log(y) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e+50], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5000000000000003e50

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.1%

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

   if 6.5000000000000003e50 < y

   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. sub-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 99.7%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   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 z around 0 88.6%

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

      1. +-commutative 88.6%

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

      2. mul-1-neg 88.6%

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

      3. +-commutative 88.6%

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

      4. *-commutative 88.6%

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

      5. sub-neg 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in y around 0 88.7%

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

      1. *-commutative 88.7%

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

      2. cancel-sign-sub-inv 88.7%

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

      3. +-commutative 88.7%

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

      4. *-commutative 88.7%

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

      5. distribute-lft-out-- 88.6%

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

      6. *-rgt-identity 88.6%

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

      7. sub-neg 88.6%

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

      8. neg-mul-1 88.6%

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

      9. associate-+r+ 88.6%

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

      10. log-rec 88.6%

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

      11. *-commutative 88.6%

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

      12. associate-+r+ 88.6%

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

      13. neg-mul-1 88.6%

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

      14. distribute-rgt-neg-in 88.6%

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

      15. log-rec 88.6%

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

      16. distribute-rgt-in 88.6%

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

      17. log-rec 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in y around inf 88.7%

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

       1. sub-neg 88.7%

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

       2. mul-1-neg 88.7%

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

       3. log-rec 88.7%

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

       4. remove-double-neg 88.7%

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

       5. metadata-eval 88.7%

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

   12. Simplified 88.7%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+50}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]

Alternative 7: 57.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-274}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-261}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.2e-274) (- x z) (if (<= y 2.9e-261) (* (log y) -0.5) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e-274) {
		tmp = x - z;
	} else if (y <= 2.9e-261) {
		tmp = log(y) * -0.5;
	} else {
		tmp = x - 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 <= 6.2d-274) then
        tmp = x - z
    else if (y <= 2.9d-261) then
        tmp = log(y) * (-0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e-274) {
		tmp = x - z;
	} else if (y <= 2.9e-261) {
		tmp = Math.log(y) * -0.5;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.2e-274:
		tmp = x - z
	elif y <= 2.9e-261:
		tmp = math.log(y) * -0.5
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e-274)
		tmp = Float64(x - z);
	elseif (y <= 2.9e-261)
		tmp = Float64(log(y) * -0.5);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.2e-274)
		tmp = x - z;
	elseif (y <= 2.9e-261)
		tmp = log(y) * -0.5;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.2e-274], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.9e-261], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.19999999999999956e-274 or 2.89999999999999985e-261 < y

   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. *-commutative 99.8%

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

      2. flip-+ 76.6%

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

      3. associate-*r/ 76.6%

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

      4. fma-neg 76.6%

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

      5. metadata-eval 76.6%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}{y - 0.5}\right) + y\right) - z \]

      6. metadata-eval 76.6%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y - 0.5}\right) + y\right) - z \]

      7. sub-neg 76.6%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      8. metadata-eval 76.6%

         \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   3. Applied egg-rr 76.6%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}\right) + y\right) - z \]
   4. Step-by-step derivation

      1. associate-/l* 76.6%

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) + y\right) - z \]

      2. associate-/r/ 76.6%

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)}\right) + y\right) - z \]

      3. +-commutative 76.6%

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{-0.5 + y}} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   5. Simplified 76.6%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{-0.5 + y} \cdot \mathsf{fma}\left(y, y, -0.25\right)}\right) + y\right) - z \]
   6. Step-by-step derivation

      1. div-inv 76.6%

         \[\leadsto \left(\left(x - \color{blue}{\left(\log y \cdot \frac{1}{-0.5 + y}\right)} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

      2. +-commutative 76.6%

         \[\leadsto \left(\left(x - \left(\log y \cdot \frac{1}{\color{blue}{y + -0.5}}\right) \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   7. Applied egg-rr 76.6%

      \[\leadsto \left(\left(x - \color{blue}{\left(\log y \cdot \frac{1}{y + -0.5}\right)} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   8. Taylor expanded in x around inf 57.4%

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

   if 6.19999999999999956e-274 < y < 2.89999999999999985e-261

   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. sub-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-sub0 100.0%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 100.0%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 100.0%

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

      9. neg-mul-1 100.0%

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

   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 z around 0 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. +-commutative 84.9%

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

      4. *-commutative 84.9%

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

      5. sub-neg 84.9%

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

   6. Simplified 84.9%

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

   7. Taylor expanded in x around 0 84.9%

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

   8. Taylor expanded in y around 0 84.9%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} \]
   9. Step-by-step derivation

      1. *-commutative 84.9%

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

   10. Simplified 84.9%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-274}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-261}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 8: 48.7% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+50}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.75e+50) x (if (<= x 6.6e+44) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.75e+50) {
		tmp = x;
	} else if (x <= 6.6e+44) {
		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 <= (-1.75d+50)) then
        tmp = x
    else if (x <= 6.6d+44) 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 <= -1.75e+50) {
		tmp = x;
	} else if (x <= 6.6e+44) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.75e+50:
		tmp = x
	elif x <= 6.6e+44:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.75e+50)
		tmp = x;
	elseif (x <= 6.6e+44)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.75e+50)
		tmp = x;
	elseif (x <= 6.6e+44)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.75e+50], x, If[LessEqual[x, 6.6e+44], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75000000000000003e50 or 6.60000000000000027e44 < 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. sub-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

      5. sub-neg 99.9%

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

      6. neg-sub0 99.9%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 99.9%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 99.9%

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

      9. neg-mul-1 99.9%

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

   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 68.2%

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

   if -1.75000000000000003e50 < x < 6.60000000000000027e44

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 77.0%

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

      1. *-commutative 77.0%

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

      2. log-rec 77.0%

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

      3. distribute-lft-neg-in 77.0%

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

      4. distribute-rgt-neg-in 77.0%

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

   4. Simplified 77.0%

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

   5. Taylor expanded in y around 0 38.7%

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

      1. neg-mul-1 38.7%

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

   7. Simplified 38.7%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+50}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 58.1% 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. *-commutative 99.8%

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

   2. flip-+ 77.2%

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

   3. associate-*r/ 77.2%

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

   4. fma-neg 77.2%

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

   5. metadata-eval 77.2%

      \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}{y - 0.5}\right) + y\right) - z \]

   6. metadata-eval 77.2%

      \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y - 0.5}\right) + y\right) - z \]

   7. sub-neg 77.2%

      \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

   8. metadata-eval 77.2%

      \[\leadsto \left(\left(x - \frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

3. Applied egg-rr 77.2%

   \[\leadsto \left(\left(x - \color{blue}{\frac{\log y \cdot \mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}\right) + y\right) - z \]
4. Step-by-step derivation

   1. associate-/l* 77.2%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) + y\right) - z \]

   2. associate-/r/ 77.2%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)}\right) + y\right) - z \]

   3. +-commutative 77.2%

      \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{-0.5 + y}} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

5. Simplified 77.2%

   \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{-0.5 + y} \cdot \mathsf{fma}\left(y, y, -0.25\right)}\right) + y\right) - z \]
6. Step-by-step derivation

   1. div-inv 77.2%

      \[\leadsto \left(\left(x - \color{blue}{\left(\log y \cdot \frac{1}{-0.5 + y}\right)} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

   2. +-commutative 77.2%

      \[\leadsto \left(\left(x - \left(\log y \cdot \frac{1}{\color{blue}{y + -0.5}}\right) \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

7. Applied egg-rr 77.2%

   \[\leadsto \left(\left(x - \color{blue}{\left(\log y \cdot \frac{1}{y + -0.5}\right)} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right) + y\right) - z \]

8. Taylor expanded in x around inf 56.3%

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

9. Final simplification 56.3%

   \[\leadsto x - z \]

Alternative 10: 29.4% 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. sub-neg 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+l+ 99.8%

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

   5. sub-neg 99.8%

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

   6. neg-sub0 99.8%

      \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

   7. associate-+l- 99.8%

      \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

   8. neg-sub0 99.8%

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

   9. neg-mul-1 99.8%

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

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 30.7%

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

5. Final simplification 30.7%

   \[\leadsto x \]

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 2023240 
(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))