Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.7s

Alternatives: 10

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: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+52} \lor \neg \left(z \leq 3.9 \cdot 10^{+58}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.9e+52) (not (<= z 3.9e+58)))
   (- x z)
   (+ x (- y (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.9e+52) || !(z <= 3.9e+58)) {
		tmp = x - z;
	} else {
		tmp = x + (y - (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 ((z <= (-2.9d+52)) .or. (.not. (z <= 3.9d+58))) then
        tmp = x - z
    else
        tmp = x + (y - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.9e+52) || !(z <= 3.9e+58)) {
		tmp = x - z;
	} else {
		tmp = x + (y - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.9e+52) or not (z <= 3.9e+58):
		tmp = x - z
	else:
		tmp = x + (y - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.9e+52) || !(z <= 3.9e+58))
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.9e+52) || ~((z <= 3.9e+58)))
		tmp = x - z;
	else
		tmp = x + (y - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.9e+52], N[Not[LessEqual[z, 3.9e+58]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x + N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e52 or 3.9000000000000001e58 < 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 x around inf 89.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{\log y \cdot \left(0.5 + y\right)}{x}\right)} + \left(y - z\right) \]
   6. 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. unsub-neg 89.6%

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

      3. associate-/l* 89.6%

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

      4. +-commutative 89.6%

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

   7. Simplified 89.6%

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

   8. Taylor expanded in y around inf 89.6%

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

      1. mul-1-neg 89.6%

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

      2. associate-/l* 89.6%

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

      3. log-rec 89.6%

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

      4. distribute-frac-neg 89.6%

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

      5. distribute-rgt-neg-in 89.6%

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

      6. associate-/l* 89.6%

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

      7. distribute-neg-frac2 89.6%

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

      8. distribute-frac-neg2 89.6%

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

      9. remove-double-neg 89.6%

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

      10. associate-/l* 89.6%

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

   10. Simplified 89.6%

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

   11. Taylor expanded in y around 0 88.1%

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

   if -2.9e52 < z < 3.9000000000000001e58

   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 z around 0 95.6%

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

      1. associate-*r* 95.6%

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

      2. neg-mul-1 95.6%

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

      3. +-commutative 95.6%

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

      4. cancel-sign-sub-inv 95.6%

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

   7. Simplified 95.6%

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

   8. Taylor expanded in y around inf 78.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+52} \lor \neg \left(z \leq 3.9 \cdot 10^{+58}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;y \leq 1.15 \cdot 10^{+78}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\left(y - z\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= y 1.15e+78)
     (- (+ x (* (log y) -0.5)) z)
     (if (<= y 1.5e+146) (- (- y z) t_0) (+ x (- y t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (y <= 1.15e+78) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if (y <= 1.5e+146) {
		tmp = (y - z) - t_0;
	} else {
		tmp = x + (y - t_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) :: t_0
    real(8) :: tmp
    t_0 = y * log(y)
    if (y <= 1.15d+78) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if (y <= 1.5d+146) then
        tmp = (y - z) - t_0
    else
        tmp = x + (y - t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log(y);
	double tmp;
	if (y <= 1.15e+78) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if (y <= 1.5e+146) {
		tmp = (y - z) - t_0;
	} else {
		tmp = x + (y - t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if y <= 1.15e+78:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif y <= 1.5e+146:
		tmp = (y - z) - t_0
	else:
		tmp = x + (y - t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (y <= 1.15e+78)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif (y <= 1.5e+146)
		tmp = Float64(Float64(y - z) - t_0);
	else
		tmp = Float64(x + Float64(y - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (y <= 1.15e+78)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif (y <= 1.5e+146)
		tmp = (y - z) - t_0;
	else
		tmp = x + (y - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.15e+78], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.5e+146], N[(N[(y - z), $MachinePrecision] - t$95$0), $MachinePrecision], N[(x + N[(y - t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.1500000000000001e78

   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 95.0%

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

   if 1.1500000000000001e78 < y < 1.50000000000000001e146

   1. Initial program 99.6%

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

      1. associate--l+ 99.7%

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

   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 82.7%

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

      1. log-rec 82.7%

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

   7. Simplified 82.7%

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

   if 1.50000000000000001e146 < 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 z around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. neg-mul-1 93.2%

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

      3. +-commutative 93.2%

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

      4. cancel-sign-sub-inv 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in y around inf 93.2%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+78}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+146}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - y \cdot \log y\\ \mathbf{if}\;y \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+145}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* y (log y)))))
   (if (<= y 4e+77)
     (- (+ x (* (log y) -0.5)) z)
     (if (<= y 7.4e+145) (- t_0 z) (+ x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y - (y * log(y));
	double tmp;
	if (y <= 4e+77) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if (y <= 7.4e+145) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_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) :: t_0
    real(8) :: tmp
    t_0 = y - (y * log(y))
    if (y <= 4d+77) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if (y <= 7.4d+145) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y - (y * Math.log(y));
	double tmp;
	if (y <= 4e+77) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if (y <= 7.4e+145) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y - (y * math.log(y))
	tmp = 0
	if y <= 4e+77:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif y <= 7.4e+145:
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(y * log(y)))
	tmp = 0.0
	if (y <= 4e+77)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif (y <= 7.4e+145)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y - (y * log(y));
	tmp = 0.0;
	if (y <= 4e+77)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif (y <= 7.4e+145)
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4e+77], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 7.4e+145], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.99999999999999993e77

   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 95.0%

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

   if 3.99999999999999993e77 < y < 7.39999999999999986e145

   1. Initial program 99.6%

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

      1. associate--l+ 99.7%

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

   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 x around inf 62.9%

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

      1. mul-1-neg 62.9%

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

      2. unsub-neg 62.9%

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

      3. associate-/l* 63.0%

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

      4. +-commutative 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in y around inf 62.9%

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

      1. mul-1-neg 62.9%

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

      2. associate-/l* 62.9%

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

      3. log-rec 62.9%

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

      4. distribute-frac-neg 62.9%

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

      5. distribute-rgt-neg-in 62.9%

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

      6. associate-/l* 62.9%

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

      7. distribute-neg-frac2 62.9%

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

      8. distribute-frac-neg2 62.9%

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

      9. remove-double-neg 62.9%

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

      10. associate-/l* 62.9%

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

   10. Simplified 62.9%

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

   11. Taylor expanded in x around 0 82.6%

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

       1. neg-mul-1 82.6%

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

       2. sub-neg 82.6%

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

   13. Simplified 82.6%

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

   if 7.39999999999999986e145 < 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 z around 0 93.2%

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

      1. associate-*r* 93.2%

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

      2. neg-mul-1 93.2%

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

      3. +-commutative 93.2%

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

      4. cancel-sign-sub-inv 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in y around inf 93.2%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+77}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+145}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+78}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.5e+78) (- (+ x (* (log y) -0.5)) z) (+ x (- y (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e+78) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = x + (y - (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.5d+78) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = x + (y - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e+78) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = x + (y - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.5e+78:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = x + (y - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.5e+78)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(x + Float64(y - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.5e+78)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = x + (y - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.5e+78], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4999999999999997e78

   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 95.0%

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

   if 5.4999999999999997e78 < 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 z around 0 85.1%

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

      1. associate-*r* 85.1%

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

      2. neg-mul-1 85.1%

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

      3. +-commutative 85.1%

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

      4. cancel-sign-sub-inv 85.1%

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

   7. Simplified 85.1%

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

   8. Taylor expanded in y around inf 85.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+78}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - y \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 7: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+153}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e+153) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e+153) {
		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 <= 8d+153) 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 <= 8e+153) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8e+153:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e+153)
		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 <= 8e+153)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e+153], 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 < 8e153

   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 x around inf 92.5%

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

      1. mul-1-neg 92.5%

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

      2. unsub-neg 92.5%

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

      3. associate-/l* 92.5%

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

      4. +-commutative 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in y around inf 76.4%

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

      1. mul-1-neg 76.4%

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

      2. associate-/l* 76.4%

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

      3. log-rec 76.4%

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

      4. distribute-frac-neg 76.4%

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

      5. distribute-rgt-neg-in 76.4%

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

      6. associate-/l* 76.4%

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

      7. distribute-neg-frac2 76.4%

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

      8. distribute-frac-neg2 76.4%

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

      9. remove-double-neg 76.4%

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

      10. associate-/l* 76.4%

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

   10. Simplified 76.4%

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

   11. Taylor expanded in y around 0 71.6%

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

   if 8e153 < 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 77.6%

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

      1. log-rec 77.6%

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

      2. sub-neg 77.6%

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

   7. Simplified 77.6%

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

Alternative 8: 49.0% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+52}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e+17) x (if (<= x 1.8e+52) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e+17) {
		tmp = x;
	} else if (x <= 1.8e+52) {
		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 <= (-1.2d+17)) then
        tmp = x
    else if (x <= 1.8d+52) 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 <= -1.2e+17) {
		tmp = x;
	} else if (x <= 1.8e+52) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.2e+17:
		tmp = x
	elif x <= 1.8e+52:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e+17)
		tmp = x;
	elseif (x <= 1.8e+52)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.2e+17)
		tmp = x;
	elseif (x <= 1.8e+52)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.2e+17], x, If[LessEqual[x, 1.8e+52], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e17 or 1.8e52 < x

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-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 61.9%

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

   if -1.2e17 < x < 1.8e52

   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+ 99.8%

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

      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- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-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 z around inf 42.1%

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

      1. neg-mul-1 42.1%

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

   7. Simplified 42.1%

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

Alternative 9: 58.7% 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. Taylor expanded in x around inf 85.2%

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

   1. mul-1-neg 85.2%

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

   2. unsub-neg 85.2%

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

   3. associate-/l* 85.2%

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

   4. +-commutative 85.2%

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

7. Simplified 85.2%

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

8. Taylor expanded in y around inf 73.9%

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

   1. mul-1-neg 73.9%

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

   2. associate-/l* 73.9%

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

   3. log-rec 73.9%

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

   4. distribute-frac-neg 73.9%

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

   5. distribute-rgt-neg-in 73.9%

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

   6. associate-/l* 73.9%

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

   7. distribute-neg-frac2 73.9%

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

   8. distribute-frac-neg2 73.9%

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

   9. remove-double-neg 73.9%

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

   10. associate-/l* 73.9%

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

10. Simplified 73.9%

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

11. Taylor expanded in y around 0 57.0%

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

Alternative 10: 30.7% 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 29.1%

   \[\leadsto \color{blue}{x} \]
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 2024129 
(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))