Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 14.5s

Alternatives: 11

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x + N[(y - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $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. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\left(\left(y + \frac{1}{2}\right) \cdot \log y\right), y\right)\right), z\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

   6. log-lowering-log.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right), y\right)\right), z\right) \]

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ t_1 := t\_0 - z\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-276}:\\ \;\;\;\;x - \left(y + 0.5\right) \cdot \log y\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+26}:\\ \;\;\;\;x + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))) (t_1 (- t_0 z)))
   (if (<= z -1.9e+19)
     t_1
     (if (<= z -1.1e-276)
       (- x (* (+ y 0.5) (log y)))
       (if (<= z 3.9e+26) (+ x t_0) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double t_1 = t_0 - z;
	double tmp;
	if (z <= -1.9e+19) {
		tmp = t_1;
	} else if (z <= -1.1e-276) {
		tmp = x - ((y + 0.5) * log(y));
	} else if (z <= 3.9e+26) {
		tmp = x + t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    t_1 = t_0 - z
    if (z <= (-1.9d+19)) then
        tmp = t_1
    else if (z <= (-1.1d-276)) then
        tmp = x - ((y + 0.5d0) * log(y))
    else if (z <= 3.9d+26) then
        tmp = x + t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double t_1 = t_0 - z;
	double tmp;
	if (z <= -1.9e+19) {
		tmp = t_1;
	} else if (z <= -1.1e-276) {
		tmp = x - ((y + 0.5) * Math.log(y));
	} else if (z <= 3.9e+26) {
		tmp = x + t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	t_1 = t_0 - z
	tmp = 0
	if z <= -1.9e+19:
		tmp = t_1
	elif z <= -1.1e-276:
		tmp = x - ((y + 0.5) * math.log(y))
	elif z <= 3.9e+26:
		tmp = x + t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	t_1 = Float64(t_0 - z)
	tmp = 0.0
	if (z <= -1.9e+19)
		tmp = t_1;
	elseif (z <= -1.1e-276)
		tmp = Float64(x - Float64(Float64(y + 0.5) * log(y)));
	elseif (z <= 3.9e+26)
		tmp = Float64(x + t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	t_1 = t_0 - z;
	tmp = 0.0;
	if (z <= -1.9e+19)
		tmp = t_1;
	elseif (z <= -1.1e-276)
		tmp = x - ((y + 0.5) * log(y));
	elseif (z <= 3.9e+26)
		tmp = x + t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - z), $MachinePrecision]}, If[LessEqual[z, -1.9e+19], t$95$1, If[LessEqual[z, -1.1e-276], N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.9e+26], N[(x + t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e19 or 3.9e26 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)\right)}, z\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(1 - \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right)\right), z\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(1 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right)\right), z\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(1 - \log y\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(1 - \log y\right)\right), z\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \log y\right)\right), z\right) \]

      6. log-lowering-log.f64 84.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(y\right)\right)\right), z\right) \]

   5. Simplified 84.9%

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

   if -1.9e19 < z < -1.0999999999999999e-276

   1. Initial program 99.8%

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

      1. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\left(\left(y + \frac{1}{2}\right) \cdot \log y\right), y\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      6. log-lowering-log.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right), y\right)\right), z\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + y\right) - \log y \cdot \left(\frac{1}{2} + y\right)} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + y\right), \color{blue}{\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\log y, \color{blue}{\left(\frac{1}{2} + y\right)}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{\frac{1}{2}} + y\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(y + \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 97.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   7. Simplified 97.3%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(y, \frac{1}{2}\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 81.4%

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

   if -1.0999999999999999e-276 < z < 3.9e26

   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. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\left(\left(y + \frac{1}{2}\right) \cdot \log y\right), y\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      6. log-lowering-log.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right), y\right)\right), z\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + y\right) - \log y \cdot \left(\frac{1}{2} + y\right)} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + y\right), \color{blue}{\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\log y, \color{blue}{\left(\frac{1}{2} + y\right)}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{\frac{1}{2}} + y\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(y + \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 97.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   7. Simplified 97.1%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \color{blue}{y}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 78.2%

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

   11. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. log-rec N/A

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

       3. log-rec N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       4. mul-1-neg N/A

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

       5. mul-1-neg N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       6. log-rec N/A

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

       7. log-rec N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       8. sub-neg N/A

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

       9. distribute-lft-out-- N/A

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

       10. *-rgt-identity N/A

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

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y - y \cdot \log y\right)}\right) \]

       12. *-rgt-identity N/A

           \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot 1 - \color{blue}{y} \cdot \log y\right)\right) \]

       13. distribute-lft-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(1 - \log y\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 - \log y\right)}\right)\right) \]

       15. --lowering--.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\log y}\right)\right)\right) \]

       16. log-lowering-log.f64 78.3%

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(y\right)\right)\right)\right) \]

   13. Simplified 78.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-276}:\\ \;\;\;\;x - \left(y + 0.5\right) \cdot \log y\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+26}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-203}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-107}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+104}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e-203)
   (- x z)
   (if (<= y 6.5e-107)
     (+ x (* (log y) -0.5))
     (if (<= y 7e+104) (- (+ x y) z) (+ x (* y (- 1.0 (log y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e-203) {
		tmp = x - z;
	} else if (y <= 6.5e-107) {
		tmp = x + (log(y) * -0.5);
	} else if (y <= 7e+104) {
		tmp = (x + y) - z;
	} else {
		tmp = x + (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 <= 9.2d-203) then
        tmp = x - z
    else if (y <= 6.5d-107) then
        tmp = x + (log(y) * (-0.5d0))
    else if (y <= 7d+104) then
        tmp = (x + y) - z
    else
        tmp = x + (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 <= 9.2e-203) {
		tmp = x - z;
	} else if (y <= 6.5e-107) {
		tmp = x + (Math.log(y) * -0.5);
	} else if (y <= 7e+104) {
		tmp = (x + y) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.2e-203:
		tmp = x - z
	elif y <= 6.5e-107:
		tmp = x + (math.log(y) * -0.5)
	elif y <= 7e+104:
		tmp = (x + y) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e-203)
		tmp = Float64(x - z);
	elseif (y <= 6.5e-107)
		tmp = Float64(x + Float64(log(y) * -0.5));
	elseif (y <= 7e+104)
		tmp = Float64(Float64(x + y) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.2e-203)
		tmp = x - z;
	elseif (y <= 6.5e-107)
		tmp = x + (log(y) * -0.5);
	elseif (y <= 7e+104)
		tmp = (x + y) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e-203], N[(x - z), $MachinePrecision], If[LessEqual[y, 6.5e-107], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+104], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 9.19999999999999966e-203

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, z\right) \]
   4. Step-by-step derivation

   5. Simplified 78.1%

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

   if 9.19999999999999966e-203 < y < 6.5000000000000002e-107

   1. Initial program 100.0%

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

      1. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\left(\left(y + \frac{1}{2}\right) \cdot \log y\right), y\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      6. log-lowering-log.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right), y\right)\right), z\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + y\right) - \log y \cdot \left(\frac{1}{2} + y\right)} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + y\right), \color{blue}{\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\log y, \color{blue}{\left(\frac{1}{2} + y\right)}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{\frac{1}{2}} + y\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(y + \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 82.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   7. Simplified 82.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - \frac{1}{2} \cdot \log y} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \log y} \]

      2. metadata-eval N/A

         \[\leadsto x + \frac{-1}{2} \cdot \log \color{blue}{y} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \log y\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\log y \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right) \]

      6. log-lowering-log.f64 82.1%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right) \]

   10. Simplified 82.1%

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

   if 6.5000000000000002e-107 < y < 7.0000000000000003e104

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\color{blue}{x}, y\right), z\right) \]
   4. Step-by-step derivation

   5. Simplified 79.5%

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

   if 7.0000000000000003e104 < y

   1. Initial program 99.6%

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

      1. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\left(\left(y + \frac{1}{2}\right) \cdot \log y\right), y\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      6. log-lowering-log.f64 99.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right), y\right)\right), z\right) \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + y\right) - \log y \cdot \left(\frac{1}{2} + y\right)} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + y\right), \color{blue}{\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\log y, \color{blue}{\left(\frac{1}{2} + y\right)}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{\frac{1}{2}} + y\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(y + \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 86.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   7. Simplified 86.0%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \color{blue}{y}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 86.0%

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

   11. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. log-rec N/A

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

       3. log-rec N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       4. mul-1-neg N/A

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

       5. mul-1-neg N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       6. log-rec N/A

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

       7. log-rec N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       8. sub-neg N/A

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

       9. distribute-lft-out-- N/A

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

       10. *-rgt-identity N/A

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

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y - y \cdot \log y\right)}\right) \]

       12. *-rgt-identity N/A

           \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot 1 - \color{blue}{y} \cdot \log y\right)\right) \]

       13. distribute-lft-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(1 - \log y\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 - \log y\right)}\right)\right) \]

       15. --lowering--.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\log y}\right)\right)\right) \]

       16. log-lowering-log.f64 86.1%

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(y\right)\right)\right)\right) \]

   13. Simplified 86.1%

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

Alternative 4: 70.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{-202}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-106}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+126}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.05e-202)
   (- x z)
   (if (<= y 2.65e-106)
     (+ x (* (log y) -0.5))
     (if (<= y 8e+126) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.05e-202) {
		tmp = x - z;
	} else if (y <= 2.65e-106) {
		tmp = x + (log(y) * -0.5);
	} else if (y <= 8e+126) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.05d-202) then
        tmp = x - z
    else if (y <= 2.65d-106) then
        tmp = x + (log(y) * (-0.5d0))
    else if (y <= 8d+126) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.05e-202) {
		tmp = x - z;
	} else if (y <= 2.65e-106) {
		tmp = x + (Math.log(y) * -0.5);
	} else if (y <= 8e+126) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.05e-202:
		tmp = x - z
	elif y <= 2.65e-106:
		tmp = x + (math.log(y) * -0.5)
	elif y <= 8e+126:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.05e-202)
		tmp = Float64(x - z);
	elseif (y <= 2.65e-106)
		tmp = Float64(x + Float64(log(y) * -0.5));
	elseif (y <= 8e+126)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.05e-202)
		tmp = x - z;
	elseif (y <= 2.65e-106)
		tmp = x + (log(y) * -0.5);
	elseif (y <= 8e+126)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.05e-202], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.65e-106], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+126], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.0500000000000002e-202 or 2.6499999999999999e-106 < y < 7.9999999999999994e126

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, z\right) \]
   4. Step-by-step derivation

   5. Simplified 78.4%

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

   if 2.0500000000000002e-202 < y < 2.6499999999999999e-106

   1. Initial program 100.0%

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

      1. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\left(\left(y + \frac{1}{2}\right) \cdot \log y\right), y\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      6. log-lowering-log.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right), y\right)\right), z\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + y\right) - \log y \cdot \left(\frac{1}{2} + y\right)} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + y\right), \color{blue}{\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\log y, \color{blue}{\left(\frac{1}{2} + y\right)}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{\frac{1}{2}} + y\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(y + \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 82.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   7. Simplified 82.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - \frac{1}{2} \cdot \log y} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \log y} \]

      2. metadata-eval N/A

         \[\leadsto x + \frac{-1}{2} \cdot \log \color{blue}{y} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \log y\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\log y \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right) \]

      6. log-lowering-log.f64 82.1%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right) \]

   10. Simplified 82.1%

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

   if 7.9999999999999994e126 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

         \[\leadsto y \cdot \left(1 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto y \cdot \left(1 - \log y\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 - \log y\right)}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\log y}\right)\right) \]

      6. log-lowering-log.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(y\right)\right)\right) \]

   5. Simplified 72.8%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y \cdot \left(1 - \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28)
   (- (+ x (* (log y) -0.5)) z)
   (- (+ x (* y (- 1.0 (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = (x + (y * (1.0 - log(y)))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.28d0) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = (x + (y * (1.0d0 - log(y)))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (x + (y * (1.0 - Math.log(y)))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.28:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (x + (y * (1.0 - math.log(y)))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.28)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 - log(y)))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.28)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (x + (y * (1.0 - log(y)))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)}, z\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right), z\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right), z\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \left(\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), z\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \left(\log y \cdot \frac{-1}{2}\right)\right), z\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\log y, \frac{-1}{2}\right)\right), z\right) \]

      7. log-lowering-log.f64 99.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right), z\right) \]

   5. Simplified 99.4%

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

   if 0.28000000000000003 < 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. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\left(\left(y + \frac{1}{2}\right) \cdot \log y\right), y\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      6. log-lowering-log.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right), y\right)\right), z\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \color{blue}{\left(y \cdot \left(-1 \cdot \log \left(\frac{1}{y}\right) - 1\right)\right)}\right), z\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right) - 1\right)\right)\right), z\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right) - 1\right)\right)\right), z\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot \left(\log y - 1\right)\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(\log y - 1\right)\right)\right), z\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(\log y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), z\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(\log y + -1\right)\right)\right), z\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\log y, -1\right)\right)\right), z\right) \]

      8. log-lowering-log.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{log.f64}\left(y\right), -1\right)\right)\right), z\right) \]

   7. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y \cdot \left(1 - \log y\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.9 \cdot 10^{+103}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.9e+103) (- (+ x (* (log y) -0.5)) z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.9e+103) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.9d+103) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.9e+103) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.9e+103:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.9e+103)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.9e+103)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.9e+103], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.8999999999999999e103

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)}, z\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right), z\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right), z\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \left(\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), z\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \left(\log y \cdot \frac{-1}{2}\right)\right), z\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\log y, \frac{-1}{2}\right)\right), z\right) \]

      7. log-lowering-log.f64 95.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right), z\right) \]

   5. Simplified 95.1%

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

   if 5.8999999999999999e103 < y

   1. Initial program 99.6%

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

      1. associate-+l- N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(y + \frac{1}{2}\right) \cdot \log y - y\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\left(\left(y + \frac{1}{2}\right) \cdot \log y\right), y\right)\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log y\right), y\right)\right), z\right) \]

      6. log-lowering-log.f64 99.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right), y\right)\right), z\right) \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + y\right) - \log y \cdot \left(\frac{1}{2} + y\right)} \]
   6. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + y\right), \color{blue}{\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\color{blue}{\log y} \cdot \left(\frac{1}{2} + y\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\log y, \color{blue}{\left(\frac{1}{2} + y\right)}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{\frac{1}{2}} + y\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(y + \color{blue}{\frac{1}{2}}\right)\right)\right) \]

      7. +-lowering-+.f64 86.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   7. Simplified 86.0%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \color{blue}{y}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 86.0%

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

   11. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. log-rec N/A

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

       3. log-rec N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       4. mul-1-neg N/A

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

       5. mul-1-neg N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       6. log-rec N/A

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

       7. log-rec N/A

          \[\leadsto x + y \cdot \left(1 + \left(\mathsf{neg}\left(\log y\right)\right)\right) \]

       8. sub-neg N/A

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

       9. distribute-lft-out-- N/A

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

       10. *-rgt-identity N/A

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

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(y - y \cdot \log y\right)}\right) \]

       12. *-rgt-identity N/A

           \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot 1 - \color{blue}{y} \cdot \log y\right)\right) \]

       13. distribute-lft-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(1 - \log y\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 - \log y\right)}\right)\right) \]

       15. --lowering--.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\log y}\right)\right)\right) \]

       16. log-lowering-log.f64 86.1%

           \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(y\right)\right)\right)\right) \]

   13. Simplified 86.1%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 8: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+128}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.6e+128) (- (+ x y) z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.6e+128) {
		tmp = (x + y) - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.6d+128) then
        tmp = (x + y) - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.6e+128) {
		tmp = (x + y) - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.6e+128:
		tmp = (x + y) - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.6e+128)
		tmp = Float64(Float64(x + y) - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.6e+128)
		tmp = (x + y) - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.6e+128], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e128

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\color{blue}{x}, y\right), z\right) \]
   4. Step-by-step derivation

   5. Simplified 73.5%

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

   if 2.6e128 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

         \[\leadsto y \cdot \left(1 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto y \cdot \left(1 - \log y\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 - \log y\right)}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\log y}\right)\right) \]

      6. log-lowering-log.f64 72.8%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(y\right)\right)\right) \]

   5. Simplified 72.8%

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

Alternative 9: 48.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+41}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 5000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.3e+41) (- 0.0 z) (if (<= z 5000000000000.0) x (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e+41) {
		tmp = 0.0 - z;
	} else if (z <= 5000000000000.0) {
		tmp = x;
	} else {
		tmp = 0.0 - 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 <= (-2.3d+41)) then
        tmp = 0.0d0 - z
    else if (z <= 5000000000000.0d0) then
        tmp = x
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e+41) {
		tmp = 0.0 - z;
	} else if (z <= 5000000000000.0) {
		tmp = x;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.3e+41:
		tmp = 0.0 - z
	elif z <= 5000000000000.0:
		tmp = x
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.3e+41)
		tmp = Float64(0.0 - z);
	elseif (z <= 5000000000000.0)
		tmp = x;
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.3e+41)
		tmp = 0.0 - z;
	elseif (z <= 5000000000000.0)
		tmp = x;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.3e+41], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 5000000000000.0], x, N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2999999999999998e41 or 5e12 < 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 z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 63.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 63.8%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-lowering-neg.f64 63.8%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 63.8%

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

   if -2.2999999999999998e41 < z < 5e12

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 39.6%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+41}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 5000000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% 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. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, z\right) \]
4. Step-by-step derivation

5. Simplified 58.3%

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

Alternative 11: 29.8% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 28.4%

   \[\leadsto \color{blue}{x} \]
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 2024191 
(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))