Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+41} \lor \neg \left(x \leq 2.8 \cdot 10^{+84}\right):\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.65e+41) (not (<= x 2.8e+84)))
   (- x (* y (+ (log y) -1.0)))
   (- (- y (* (log y) (+ y 0.5))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.65e+41) || !(x <= 2.8e+84)) {
		tmp = x - (y * (log(y) + -1.0));
	} else {
		tmp = (y - (log(y) * (y + 0.5))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.65d+41)) .or. (.not. (x <= 2.8d+84))) then
        tmp = x - (y * (log(y) + (-1.0d0)))
    else
        tmp = (y - (log(y) * (y + 0.5d0))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.65e+41) || !(x <= 2.8e+84)) {
		tmp = x - (y * (Math.log(y) + -1.0));
	} else {
		tmp = (y - (Math.log(y) * (y + 0.5))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.65e+41) or not (x <= 2.8e+84):
		tmp = x - (y * (math.log(y) + -1.0))
	else:
		tmp = (y - (math.log(y) * (y + 0.5))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.65e+41) || !(x <= 2.8e+84))
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	else
		tmp = Float64(Float64(y - Float64(log(y) * Float64(y + 0.5))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.65e+41) || ~((x <= 2.8e+84)))
		tmp = x - (y * (log(y) + -1.0));
	else
		tmp = (y - (log(y) * (y + 0.5))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.65e+41], N[Not[LessEqual[x, 2.8e+84]], $MachinePrecision]], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e41 or 2.79999999999999982e84 < x

   1. Initial program 99.9%

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

      1. add-exp-log 59.0%

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

   4. Applied egg-rr 59.0%

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

   5. Taylor expanded in z around 0 92.4%

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

      1. distribute-rgt-in 92.4%

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

      2. +-commutative 92.4%

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

      3. +-commutative 92.4%

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

      4. distribute-rgt-in 92.4%

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

      5. associate-+r- 92.4%

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

      6. +-commutative 92.4%

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

      7. associate--r- 92.5%

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

      8. distribute-rgt-in 92.5%

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

      9. +-commutative 92.5%

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

      10. distribute-rgt-in 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in y around inf 92.5%

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

      1. sub-neg 92.5%

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

      2. mul-1-neg 92.5%

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

      3. log-rec 92.5%

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

      4. remove-double-neg 92.5%

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

      5. metadata-eval 92.5%

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

   10. Simplified 92.5%

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

   if -1.65e41 < x < 2.79999999999999982e84

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-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 0 96.9%

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

      1. associate-*r* 96.9%

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

      2. neg-mul-1 96.9%

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

      3. +-commutative 96.9%

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

      4. cancel-sign-sub-inv 96.9%

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

   7. Simplified 96.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+41} \lor \neg \left(x \leq 2.8 \cdot 10^{+84}\right):\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+27}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+29}:\\ \;\;\;\;x + \left(y - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e+27)
   (- x z)
   (if (<= z 2.9e+29)
     (+ x (- y (* (log y) (+ y 0.5))))
     (- (- y (* y (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+27) {
		tmp = x - z;
	} else if (z <= 2.9e+29) {
		tmp = x + (y - (log(y) * (y + 0.5)));
	} else {
		tmp = (y - (y * log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.1d+27)) then
        tmp = x - z
    else if (z <= 2.9d+29) then
        tmp = x + (y - (log(y) * (y + 0.5d0)))
    else
        tmp = (y - (y * log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e+27) {
		tmp = x - z;
	} else if (z <= 2.9e+29) {
		tmp = x + (y - (Math.log(y) * (y + 0.5)));
	} else {
		tmp = (y - (y * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e+27:
		tmp = x - z
	elif z <= 2.9e+29:
		tmp = x + (y - (math.log(y) * (y + 0.5)))
	else:
		tmp = (y - (y * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e+27)
		tmp = Float64(x - z);
	elseif (z <= 2.9e+29)
		tmp = Float64(x + Float64(y - Float64(log(y) * Float64(y + 0.5))));
	else
		tmp = Float64(Float64(y - Float64(y * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.1e+27)
		tmp = x - z;
	elseif (z <= 2.9e+29)
		tmp = x + (y - (log(y) * (y + 0.5)));
	else
		tmp = (y - (y * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e+27], N[(x - z), $MachinePrecision], If[LessEqual[z, 2.9e+29], N[(x + N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.09999999999999995e27

   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 99.9%

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

   6. Taylor expanded in y around inf 99.9%

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

      1. log-rec 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. unsub-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around 0 82.3%

      \[\leadsto x + z \cdot \color{blue}{-1} \]

   if -2.09999999999999995e27 < z < 2.8999999999999999e29

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-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 97.9%

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

      1. associate-*r* 97.9%

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

      2. neg-mul-1 97.9%

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

      3. +-commutative 97.9%

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

      4. cancel-sign-sub-inv 97.9%

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

   7. Simplified 97.9%

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

   if 2.8999999999999999e29 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 90.6%

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

      1. log-rec 90.6%

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

   5. Simplified 90.6%

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

      1. add-exp-log 62.7%

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

   7. Applied egg-rr 55.4%

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

      1. +-commutative 55.4%

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

      2. rem-exp-log 90.6%

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

      3. distribute-rgt-neg-out 90.6%

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

      4. unsub-neg 90.6%

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

   9. Applied egg-rr 90.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+27}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+29}:\\ \;\;\;\;x + \left(y - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+53} \lor \neg \left(x \leq 1.1 \cdot 10^{+84}\right):\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.25e+53) (not (<= x 1.1e+84)))
   (- x (* y (+ (log y) -1.0)))
   (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e+53) || !(x <= 1.1e+84)) {
		tmp = x - (y * (log(y) + -1.0));
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.25d+53)) .or. (.not. (x <= 1.1d+84))) then
        tmp = x - (y * (log(y) + (-1.0d0)))
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e+53) || !(x <= 1.1e+84)) {
		tmp = x - (y * (Math.log(y) + -1.0));
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.25e+53) or not (x <= 1.1e+84):
		tmp = x - (y * (math.log(y) + -1.0))
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.25e+53) || !(x <= 1.1e+84))
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.25e+53) || ~((x <= 1.1e+84)))
		tmp = x - (y * (log(y) + -1.0));
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.25e+53], N[Not[LessEqual[x, 1.1e+84]], $MachinePrecision]], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2500000000000001e53 or 1.0999999999999999e84 < x

   1. Initial program 99.9%

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

      1. add-exp-log 59.0%

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

   4. Applied egg-rr 59.0%

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

   5. Taylor expanded in z around 0 92.4%

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

      1. distribute-rgt-in 92.4%

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

      2. +-commutative 92.4%

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

      3. +-commutative 92.4%

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

      4. distribute-rgt-in 92.4%

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

      5. associate-+r- 92.4%

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

      6. +-commutative 92.4%

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

      7. associate--r- 92.5%

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

      8. distribute-rgt-in 92.5%

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

      9. +-commutative 92.5%

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

      10. distribute-rgt-in 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in y around inf 92.5%

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

      1. sub-neg 92.5%

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

      2. mul-1-neg 92.5%

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

      3. log-rec 92.5%

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

      4. remove-double-neg 92.5%

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

      5. metadata-eval 92.5%

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

   10. Simplified 92.5%

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

   if -1.2500000000000001e53 < x < 1.0999999999999999e84

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf 80.9%

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

      1. log-rec 80.9%

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

   5. Simplified 80.9%

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

   6. Taylor expanded in y around 0 80.9%

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

      1. neg-mul-1 80.9%

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

      2. log-rec 80.9%

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

      3. log-rec 80.9%

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

      4. sub-neg 80.9%

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

   8. Simplified 80.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+53} \lor \neg \left(x \leq 1.1 \cdot 10^{+84}\right):\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+36}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e+36)
   (- (+ x (* (log y) -0.5)) z)
   (- x (* y (+ (log y) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e+36) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = x - (y * (log(y) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.25d+36) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = x - (y * (log(y) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e+36) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = x - (y * (Math.log(y) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.25e+36:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = x - (y * (math.log(y) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.25e+36)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.25e+36)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = x - (y * (log(y) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.25e+36], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.24999999999999994e36

   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 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 96.3%

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

   if 1.24999999999999994e36 < y

   1. Initial program 99.7%

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

      1. add-exp-log 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in z around 0 84.8%

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

      1. distribute-rgt-in 84.8%

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

      2. +-commutative 84.8%

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

      3. +-commutative 84.8%

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

      4. distribute-rgt-in 84.8%

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

      5. associate-+r- 84.8%

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

      6. +-commutative 84.8%

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

      7. associate--r- 84.8%

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

      8. distribute-rgt-in 84.8%

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

      9. +-commutative 84.8%

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

      10. distribute-rgt-in 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in y around inf 84.9%

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

      1. sub-neg 84.9%

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

      2. mul-1-neg 84.9%

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

      3. log-rec 84.9%

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

      4. remove-double-neg 84.9%

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

      5. metadata-eval 84.9%

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

   10. Simplified 84.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+36}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+34}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.85e+34) (- x z) (- x (* y (+ (log y) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.85e+34) {
		tmp = x - z;
	} else {
		tmp = x - (y * (log(y) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.85d+34) then
        tmp = x - z
    else
        tmp = x - (y * (log(y) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.85e+34) {
		tmp = x - z;
	} else {
		tmp = x - (y * (Math.log(y) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.85e+34:
		tmp = x - z
	else:
		tmp = x - (y * (math.log(y) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.85e+34)
		tmp = Float64(x - z);
	else
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.85e+34)
		tmp = x - z;
	else
		tmp = x - (y * (log(y) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.85e+34], N[(x - z), $MachinePrecision], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.84999999999999987e34

   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 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 z around inf 99.9%

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

   6. Taylor expanded in y around inf 74.9%

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

      1. log-rec 74.9%

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

      2. distribute-frac-neg 74.9%

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

      3. unsub-neg 74.9%

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

   8. Simplified 74.9%

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

   9. Taylor expanded in y around 0 72.4%

      \[\leadsto x + z \cdot \color{blue}{-1} \]

   if 2.84999999999999987e34 < y

   1. Initial program 99.7%

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

      1. add-exp-log 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in z around 0 84.8%

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

      1. distribute-rgt-in 84.8%

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

      2. +-commutative 84.8%

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

      3. +-commutative 84.8%

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

      4. distribute-rgt-in 84.8%

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

      5. associate-+r- 84.8%

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

      6. +-commutative 84.8%

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

      7. associate--r- 84.8%

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

      8. distribute-rgt-in 84.8%

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

      9. +-commutative 84.8%

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

      10. distribute-rgt-in 84.8%

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

   7. Simplified 84.8%

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

   8. Taylor expanded in y around inf 84.9%

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

      1. sub-neg 84.9%

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

      2. mul-1-neg 84.9%

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

      3. log-rec 84.9%

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

      4. remove-double-neg 84.9%

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

      5. metadata-eval 84.9%

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

   10. Simplified 84.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+34}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate--l+ 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+167}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.7e+167) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e+167) {
		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 <= 1.7d+167) 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 <= 1.7e+167) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.7e+167:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.7e+167)
		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 <= 1.7e+167)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.7e+167], 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 < 1.7e167

   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+ 99.8%

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

      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+ 99.9%

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

      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 92.4%

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

   6. Taylor expanded in y around inf 76.3%

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

      1. log-rec 76.3%

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

      2. distribute-frac-neg 76.3%

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

      3. unsub-neg 76.3%

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

   8. Simplified 76.3%

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

   9. Taylor expanded in y around 0 66.9%

      \[\leadsto x + z \cdot \color{blue}{-1} \]

   if 1.7e167 < 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+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-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 82.3%

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

      1. log-rec 82.3%

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

      2. sub-neg 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+167}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.1% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+82}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e+48) x (if (<= x 1.3e+82) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e+48) {
		tmp = x;
	} else if (x <= 1.3e+82) {
		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+48)) then
        tmp = x
    else if (x <= 1.3d+82) 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+48) {
		tmp = x;
	} else if (x <= 1.3e+82) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.2e+48:
		tmp = x
	elif x <= 1.3e+82:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.2e+48)
		tmp = x;
	elseif (x <= 1.3e+82)
		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+48)
		tmp = x;
	elseif (x <= 1.3e+82)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.2e+48], x, If[LessEqual[x, 1.3e+82], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2000000000000001e48 or 1.2999999999999999e82 < 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+ 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 x around inf 78.0%

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

   if -1.2000000000000001e48 < x < 1.2999999999999999e82

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-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 36.6%

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

      1. neg-mul-1 36.6%

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

   7. Simplified 36.6%

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

Alternative 11: 58.1% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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 z around inf 80.7%

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

6. Taylor expanded in y around inf 70.5%

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

   1. log-rec 70.5%

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

   2. distribute-frac-neg 70.5%

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

   3. unsub-neg 70.5%

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

8. Simplified 70.5%

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

9. Taylor expanded in y around 0 53.1%

   \[\leadsto x + z \cdot \color{blue}{-1} \]

10. Final simplification 53.1%

    \[\leadsto x - z \]
11. Add Preprocessing

Alternative 12: 29.9% 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 27.4%

   \[\leadsto \color{blue}{x} \]
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 2024144 
(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))