Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.2s

Alternatives: 12

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-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-184}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-127}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 8.1 \cdot 10^{+102}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.3e-184)
   (- x z)
   (if (<= y 2.75e-127)
     (+ x (* (log y) -0.5))
     (if (<= y 8.1e+102) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e-184) {
		tmp = x - z;
	} else if (y <= 2.75e-127) {
		tmp = x + (log(y) * -0.5);
	} else if (y <= 8.1e+102) {
		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 <= 4.3d-184) then
        tmp = x - z
    else if (y <= 2.75d-127) then
        tmp = x + (log(y) * (-0.5d0))
    else if (y <= 8.1d+102) 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 <= 4.3e-184) {
		tmp = x - z;
	} else if (y <= 2.75e-127) {
		tmp = x + (Math.log(y) * -0.5);
	} else if (y <= 8.1e+102) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.3e-184:
		tmp = x - z
	elif y <= 2.75e-127:
		tmp = x + (math.log(y) * -0.5)
	elif y <= 8.1e+102:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.3e-184)
		tmp = Float64(x - z);
	elseif (y <= 2.75e-127)
		tmp = Float64(x + Float64(log(y) * -0.5));
	elseif (y <= 8.1e+102)
		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 <= 4.3e-184)
		tmp = x - z;
	elseif (y <= 2.75e-127)
		tmp = x + (log(y) * -0.5);
	elseif (y <= 8.1e+102)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.3e-184], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.75e-127], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.1e+102], 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 < 4.30000000000000007e-184 or 2.75000000000000018e-127 < y < 8.10000000000000037e102

   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%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 99.6%

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

      2. pow2 99.6%

         \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

   7. Taylor expanded in x around inf 96.2%

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

      1. mul-1-neg 96.2%

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

      2. associate-/l* 96.2%

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

      3. +-commutative 96.2%

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

      4. unsub-neg 96.2%

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

      5. associate-*r/ 96.2%

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

      6. *-commutative 96.2%

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

      7. associate-*r/ 96.2%

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

      8. *-commutative 96.2%

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

      9. +-commutative 96.2%

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

   9. Simplified 96.2%

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

   10. Taylor expanded in y around inf 81.2%

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

   11. Taylor expanded in y around 0 74.9%

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

   if 4.30000000000000007e-184 < y < 2.75000000000000018e-127

   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-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in z around 0 77.8%

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

   if 8.10000000000000037e102 < 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-define 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. Add Preprocessing

   5. Taylor expanded in y around inf 74.4%

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

      1. log-rec 74.4%

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

      2. sub-neg 74.4%

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

   7. Simplified 74.4%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-184}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-127}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 8.1 \cdot 10^{+102}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -400000000 \lor \neg \left(z \leq 2.4 \cdot 10^{+24}\right):\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \log y \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -400000000.0) (not (<= z 2.4e+24)))
   (- (- y z) (* y (log y)))
   (- (+ x y) (* (log y) (+ y 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -400000000.0) || !(z <= 2.4e+24)) {
		tmp = (y - z) - (y * log(y));
	} else {
		tmp = (x + y) - (log(y) * (y + 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-400000000.0d0)) .or. (.not. (z <= 2.4d+24))) then
        tmp = (y - z) - (y * log(y))
    else
        tmp = (x + y) - (log(y) * (y + 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -400000000.0) || !(z <= 2.4e+24)) {
		tmp = (y - z) - (y * Math.log(y));
	} else {
		tmp = (x + y) - (Math.log(y) * (y + 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -400000000.0) or not (z <= 2.4e+24):
		tmp = (y - z) - (y * math.log(y))
	else:
		tmp = (x + y) - (math.log(y) * (y + 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -400000000.0) || !(z <= 2.4e+24))
		tmp = Float64(Float64(y - z) - Float64(y * log(y)));
	else
		tmp = Float64(Float64(x + y) - Float64(log(y) * Float64(y + 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -400000000.0) || ~((z <= 2.4e+24)))
		tmp = (y - z) - (y * log(y));
	else
		tmp = (x + y) - (log(y) * (y + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -400000000.0], N[Not[LessEqual[z, 2.4e+24]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4e8 or 2.4000000000000001e24 < z

   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%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 88.2%

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

      1. log-rec 88.2%

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

   7. Simplified 88.2%

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

   8. Taylor expanded in z around 0 88.2%

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

      1. neg-mul-1 88.2%

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

      2. associate-+r+ 88.2%

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

      3. sub-neg 88.2%

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

      4. mul-1-neg 88.2%

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

      5. unsub-neg 88.2%

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

   10. Simplified 88.2%

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

   if -4e8 < z < 2.4000000000000001e24

   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. Add Preprocessing

   5. Taylor expanded in z around 0 98.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -400000000 \lor \neg \left(z \leq 2.4 \cdot 10^{+24}\right):\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \log y \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 52:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 - \log y\right) + \frac{x - z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 52.0)
   (- (+ x (* (log y) -0.5)) z)
   (* y (+ (- 1.0 (log y)) (/ (- x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 52.0) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = y * ((1.0 - log(y)) + ((x - z) / 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 <= 52.0d0) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = y * ((1.0d0 - log(y)) + ((x - z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 52.0) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = y * ((1.0 - Math.log(y)) + ((x - z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 52.0:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = y * ((1.0 - math.log(y)) + ((x - z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 52.0)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(y * Float64(Float64(1.0 - log(y)) + Float64(Float64(x - z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 52.0)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = y * ((1.0 - log(y)) + ((x - z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 52.0], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 52

   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-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 99.5%

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

   if 52 < 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. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

         \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

   7. Taylor expanded in x around inf 78.8%

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

      1. mul-1-neg 78.8%

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

      2. associate-/l* 78.8%

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

      3. +-commutative 78.8%

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

      4. unsub-neg 78.8%

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

      5. associate-*r/ 78.8%

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

      6. *-commutative 78.8%

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

      7. associate-*r/ 78.7%

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

      8. *-commutative 78.7%

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

      9. +-commutative 78.7%

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

   9. Simplified 78.7%

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

   10. Taylor expanded in y around inf 78.3%

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

   11. Taylor expanded in y around inf 99.2%

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

       1. log-rec 99.2%

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

       2. associate-+r+ 99.2%

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

       3. sub-neg 99.2%

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

       4. associate--l+ 99.2%

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

       5. div-sub 99.2%

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

   13. Simplified 99.2%

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

4. Final simplification 99.3%

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

Alternative 5: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-128}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+103}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e-128)
   (- (* (log y) -0.5) z)
   (if (<= y 1.42e+103) (- x z) (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e-128) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.42e+103) {
		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 <= 2.8d-128) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.42d+103) 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 <= 2.8e-128) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.42e+103) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.8e-128:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.42e+103:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e-128)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.42e+103)
		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 <= 2.8e-128)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.42e+103)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.8e-128], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.42e+103], 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 < 2.7999999999999998e-128

   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-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in x around 0 77.6%

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

      1. *-commutative 77.6%

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

   8. Simplified 77.6%

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

   if 2.7999999999999998e-128 < y < 1.42e103

   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%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 99.6%

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

      2. pow2 99.6%

         \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

   7. Taylor expanded in x around inf 94.4%

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

      1. mul-1-neg 94.4%

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

      2. associate-/l* 94.4%

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

      3. +-commutative 94.4%

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

      4. unsub-neg 94.4%

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

      5. associate-*r/ 94.4%

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

      6. *-commutative 94.4%

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

      7. associate-*r/ 94.4%

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

      8. *-commutative 94.4%

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

      9. +-commutative 94.4%

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

   9. Simplified 94.4%

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

   10. Taylor expanded in y around inf 81.1%

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

   11. Taylor expanded in y around 0 71.8%

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

   if 1.42e103 < 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-define 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. Add Preprocessing

   5. Taylor expanded in y around inf 74.4%

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

      1. log-rec 74.4%

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

      2. sub-neg 74.4%

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

   7. Simplified 74.4%

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

Alternative 6: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+19}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+19) (- (+ x (* (log y) -0.5)) z) (- (- y z) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+19) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = (y - z) - (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 <= 6d+19) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = (y - z) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+19) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (y - z) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+19:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (y - z) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+19)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(Float64(y - z) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+19)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (y - z) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+19], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6e19

   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-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 98.9%

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

   if 6e19 < 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. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 81.2%

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

      1. log-rec 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in z around 0 81.2%

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

      1. neg-mul-1 81.2%

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

      2. associate-+r+ 81.2%

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

      3. sub-neg 81.2%

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

      4. mul-1-neg 81.2%

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

      5. unsub-neg 81.2%

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

   10. Simplified 81.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+19}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{+102}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.3e+102) (- (+ x (* (log y) -0.5)) z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.3e+102) {
		tmp = (x + (log(y) * -0.5)) - 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 <= 7.3d+102) then
        tmp = (x + (log(y) * (-0.5d0))) - 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 <= 7.3e+102) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.3e+102:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.3e+102)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - 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 <= 7.3e+102)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.3e+102], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.29999999999999989e102

   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-define 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. Add Preprocessing

   5. Taylor expanded in y around 0 91.0%

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

   if 7.29999999999999989e102 < 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-define 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. Add Preprocessing

   5. Taylor expanded in y around inf 74.4%

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

      1. log-rec 74.4%

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

      2. sub-neg 74.4%

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

   7. Simplified 74.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{+102}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 9: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{+102}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.3e+102) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.3e+102) {
		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 <= 5.3d+102) 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 <= 5.3e+102) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.3e+102:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.3e+102)
		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 <= 5.3e+102)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.3e+102], 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 < 5.2999999999999997e102

   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%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 99.5%

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

      2. pow2 99.5%

         \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

   7. Taylor expanded in x around inf 96.7%

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

      1. mul-1-neg 96.7%

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

      2. associate-/l* 96.7%

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

      3. +-commutative 96.7%

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

      4. unsub-neg 96.7%

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

      5. associate-*r/ 96.7%

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

      6. *-commutative 96.7%

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

      7. associate-*r/ 96.7%

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

      8. *-commutative 96.7%

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

      9. +-commutative 96.7%

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

   9. Simplified 96.7%

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

   10. Taylor expanded in y around inf 75.7%

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

   11. Taylor expanded in y around 0 70.2%

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

   if 5.2999999999999997e102 < 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-define 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. Add Preprocessing

   5. Taylor expanded in y around inf 74.4%

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

      1. log-rec 74.4%

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

      2. sub-neg 74.4%

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

   7. Simplified 74.4%

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

Alternative 10: 47.4% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2700000 \lor \neg \left(z \leq 2.45 \cdot 10^{+20}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2700000.0) (not (<= z 2.45e+20))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2700000.0) || !(z <= 2.45e+20)) {
		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 ((z <= (-2700000.0d0)) .or. (.not. (z <= 2.45d+20))) 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 ((z <= -2700000.0) || !(z <= 2.45e+20)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2700000.0) or not (z <= 2.45e+20):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2700000.0) || !(z <= 2.45e+20))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2700000.0) || ~((z <= 2.45e+20)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2700000.0], N[Not[LessEqual[z, 2.45e+20]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -2.7e6 or 2.45e20 < z

   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-define 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. Add Preprocessing

   5. Taylor expanded in z around inf 69.1%

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

      1. neg-mul-1 69.1%

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

   7. Simplified 69.1%

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

   if -2.7e6 < z < 2.45e20

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 37.1%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2700000 \lor \neg \left(z \leq 2.45 \cdot 10^{+20}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.3% 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%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 99.4%

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

   2. pow2 99.4%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

6. Applied egg-rr 99.4%

   \[\leadsto \left(x - \color{blue}{{\left(\sqrt{y + 0.5}\right)}^{2}} \cdot \log y\right) + \left(y - z\right) \]

7. Taylor expanded in x around inf 89.6%

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

   1. mul-1-neg 89.6%

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

   2. associate-/l* 89.6%

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

   3. +-commutative 89.6%

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

   4. unsub-neg 89.6%

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

   5. associate-*r/ 89.6%

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

   6. *-commutative 89.6%

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

   7. associate-*r/ 89.6%

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

   8. *-commutative 89.6%

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

   9. +-commutative 89.6%

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

9. Simplified 89.6%

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

10. Taylor expanded in y around inf 75.0%

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

11. Taylor expanded in y around 0 56.5%

    \[\leadsto \color{blue}{x - z} \]
12. Add Preprocessing

Alternative 12: 29.8% 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-define 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. Add Preprocessing

5. Taylor expanded in x around inf 26.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 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 2024151 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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