Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 11.4s

Alternatives: 14

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: 74.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + 0.5\right) \cdot \log y\\ \mathbf{if}\;z \leq -1400000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-119}:\\ \;\;\;\;x - t\_0\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+34}:\\ \;\;\;\;y - t\_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y 0.5) (log y))))
   (if (<= z -1400000.0)
     (- x z)
     (if (<= z 3.3e-119) (- x t_0) (if (<= z 3.4e+34) (- y t_0) (- x z))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + 0.5) * log(y);
	double tmp;
	if (z <= -1400000.0) {
		tmp = x - z;
	} else if (z <= 3.3e-119) {
		tmp = x - t_0;
	} else if (z <= 3.4e+34) {
		tmp = y - t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + 0.5d0) * log(y)
    if (z <= (-1400000.0d0)) then
        tmp = x - z
    else if (z <= 3.3d-119) then
        tmp = x - t_0
    else if (z <= 3.4d+34) then
        tmp = y - t_0
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + 0.5) * Math.log(y);
	double tmp;
	if (z <= -1400000.0) {
		tmp = x - z;
	} else if (z <= 3.3e-119) {
		tmp = x - t_0;
	} else if (z <= 3.4e+34) {
		tmp = y - t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + 0.5) * math.log(y)
	tmp = 0
	if z <= -1400000.0:
		tmp = x - z
	elif z <= 3.3e-119:
		tmp = x - t_0
	elif z <= 3.4e+34:
		tmp = y - t_0
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + 0.5) * log(y))
	tmp = 0.0
	if (z <= -1400000.0)
		tmp = Float64(x - z);
	elseif (z <= 3.3e-119)
		tmp = Float64(x - t_0);
	elseif (z <= 3.4e+34)
		tmp = Float64(y - t_0);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + 0.5) * log(y);
	tmp = 0.0;
	if (z <= -1400000.0)
		tmp = x - z;
	elseif (z <= 3.3e-119)
		tmp = x - t_0;
	elseif (z <= 3.4e+34)
		tmp = y - t_0;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1400000.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 3.3e-119], N[(x - t$95$0), $MachinePrecision], If[LessEqual[z, 3.4e+34], N[(y - t$95$0), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e6 or 3.3999999999999999e34 < z

   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 79.3%

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

   if -1.4e6 < z < 3.30000000000000008e-119

   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+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. --lowering--.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

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

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

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

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

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

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

      17. log-lowering-log.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 74.4%

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

   if 3.30000000000000008e-119 < z < 3.3999999999999999e34

   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+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. --lowering--.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

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

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

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

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

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

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

      17. log-lowering-log.f64 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 75.6%

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

Alternative 3: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+78}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+113}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.8e+78)
   (- x z)
   (if (<= x 4e+113) (- (* y (- 1.0 (log y))) z) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.8e+78) {
		tmp = x - z;
	} else if (x <= 4e+113) {
		tmp = (y * (1.0 - log(y))) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-6.8d+78)) then
        tmp = x - z
    else if (x <= 4d+113) then
        tmp = (y * (1.0d0 - log(y))) - z
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.8e+78) {
		tmp = x - z;
	} else if (x <= 4e+113) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.8e+78:
		tmp = x - z
	elif x <= 4e+113:
		tmp = (y * (1.0 - math.log(y))) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.8e+78)
		tmp = Float64(x - z);
	elseif (x <= 4e+113)
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.8e+78)
		tmp = x - z;
	elseif (x <= 4e+113)
		tmp = (y * (1.0 - log(y))) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.8e+78], N[(x - z), $MachinePrecision], If[LessEqual[x, 4e+113], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.80000000000000014e78 or 4e113 < 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. Taylor expanded in x around inf

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

   5. Simplified 84.7%

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

   if -6.80000000000000014e78 < x < 4e113

   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 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 74.3%

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

   5. Simplified 74.3%

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

Alternative 4: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1820000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+15}:\\ \;\;\;\;x - \left(y + 0.5\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1820000.0)
   (- x z)
   (if (<= z 1.65e+15) (- x (* (+ y 0.5) (log y))) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1820000.0) {
		tmp = x - z;
	} else if (z <= 1.65e+15) {
		tmp = x - ((y + 0.5) * log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1820000.0d0)) then
        tmp = x - z
    else if (z <= 1.65d+15) then
        tmp = x - ((y + 0.5d0) * log(y))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1820000.0) {
		tmp = x - z;
	} else if (z <= 1.65e+15) {
		tmp = x - ((y + 0.5) * Math.log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1820000.0:
		tmp = x - z
	elif z <= 1.65e+15:
		tmp = x - ((y + 0.5) * math.log(y))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1820000.0)
		tmp = Float64(x - z);
	elseif (z <= 1.65e+15)
		tmp = Float64(x - Float64(Float64(y + 0.5) * log(y)));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1820000.0)
		tmp = x - z;
	elseif (z <= 1.65e+15)
		tmp = x - ((y + 0.5) * log(y));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1820000.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 1.65e+15], N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.82e6 or 1.65e15 < z

   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 78.2%

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

   if -1.82e6 < z < 1.65e15

   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+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. --lowering--.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

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

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

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

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

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

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

      17. log-lowering-log.f64 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 71.7%

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

Alternative 5: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -122:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+35}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -122.0) (- x z) (if (<= x 5.3e+35) (- (* (log y) -0.5) z) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -122.0) {
		tmp = x - z;
	} else if (x <= 5.3e+35) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-122.0d0)) then
        tmp = x - z
    else if (x <= 5.3d+35) then
        tmp = (log(y) * (-0.5d0)) - z
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -122.0) {
		tmp = x - z;
	} else if (x <= 5.3e+35) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -122.0:
		tmp = x - z
	elif x <= 5.3e+35:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -122.0)
		tmp = Float64(x - z);
	elseif (x <= 5.3e+35)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -122.0)
		tmp = x - z;
	elseif (x <= 5.3e+35)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -122.0], N[(x - z), $MachinePrecision], If[LessEqual[x, 5.3e+35], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -122 or 5.30000000000000009e35 < 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. Taylor expanded in x around inf

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

   5. Simplified 74.3%

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

   if -122 < x < 5.30000000000000009e35

   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 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 67.8%

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

   5. Simplified 67.8%

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

   6. Taylor expanded in x around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. log-lowering-log.f64 67.1%

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

   8. Simplified 67.1%

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

Alternative 6: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -260:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 200:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -260.0) (- x z) (if (<= z 200.0) (+ x (* (log y) -0.5)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -260.0) {
		tmp = x - z;
	} else if (z <= 200.0) {
		tmp = x + (log(y) * -0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-260.0d0)) then
        tmp = x - z
    else if (z <= 200.0d0) then
        tmp = x + (log(y) * (-0.5d0))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -260.0) {
		tmp = x - z;
	} else if (z <= 200.0) {
		tmp = x + (Math.log(y) * -0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -260.0:
		tmp = x - z
	elif z <= 200.0:
		tmp = x + (math.log(y) * -0.5)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -260.0)
		tmp = Float64(x - z);
	elseif (z <= 200.0)
		tmp = Float64(x + Float64(log(y) * -0.5));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -260.0)
		tmp = x - z;
	elseif (z <= 200.0)
		tmp = x + (log(y) * -0.5);
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -260.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 200.0], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -260 or 200 < z

   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 76.6%

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

   if -260 < z < 200

   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 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 63.2%

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

   5. Simplified 63.2%

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

   6. Taylor expanded in z around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. log-lowering-log.f64 62.7%

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

   8. Simplified 62.7%

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

Alternative 7: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(y - z\right)\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-7) (- (+ x (* (log y) -0.5)) z) (- (+ x (- y z)) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-7) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = (x + (y - z)) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8d-7) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = (x + (y - z)) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-7) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (x + (y - z)) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8e-7:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (x + (y - z)) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e-7)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(Float64(x + Float64(y - z)) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8e-7)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (x + (y - z)) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e-7], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999996e-7

   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 99.9%

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

   5. Simplified 99.9%

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

   if 7.9999999999999996e-7 < 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+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. --lowering--.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

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

      12. +-commutative N/A

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

      13. sub-neg N/A

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

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

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

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

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

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

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

      17. log-lowering-log.f64 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(y - z\right)\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-7) (- (+ x (* (log y) -0.5)) z) (- (+ y (- x (* y (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-7) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = (y + (x - (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 (y <= 8d-7) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = (y + (x - (y * log(y)))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-7) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (y + (x - (y * Math.log(y)))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8e-7:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (y + (x - (y * math.log(y)))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e-7)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(Float64(y + Float64(x - Float64(y * log(y)))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8e-7)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (y + (x - (y * log(y)))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e-7], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999996e-7

   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 99.9%

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

   5. Simplified 99.9%

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

   if 7.9999999999999996e-7 < 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. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+35}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.2e+35) (- (+ x (* (log y) -0.5)) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e+35) {
		tmp = (x + (log(y) * -0.5)) - z;
	} 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 (y <= 6.2d+35) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    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 (y <= 6.2e+35) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.2e+35:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e+35)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	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 (y <= 6.2e+35)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.2e+35], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.19999999999999973e35

   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 98.0%

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

   5. Simplified 98.0%

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

   if 6.19999999999999973e35 < 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. 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 79.4%

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

   5. Simplified 79.4%

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

Alternative 10: 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 11: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.86 \cdot 10^{+124}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.86e+124) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.86e+124) {
		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 <= 1.86d+124) 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 <= 1.86e+124) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.86e+124:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.86e+124)
		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 <= 1.86e+124)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.86e+124], 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 < 1.8599999999999999e124

   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 x around inf

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

   5. Simplified 67.1%

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

   if 1.8599999999999999e124 < 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 68.5%

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

   5. Simplified 68.5%

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

Alternative 12: 48.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+66}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7e+74) x (if (<= x 1.25e+66) (- 0.0 z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e+74) {
		tmp = x;
	} else if (x <= 1.25e+66) {
		tmp = 0.0 - 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 <= (-7d+74)) then
        tmp = x
    else if (x <= 1.25d+66) then
        tmp = 0.0d0 - 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 <= -7e+74) {
		tmp = x;
	} else if (x <= 1.25e+66) {
		tmp = 0.0 - z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7e+74:
		tmp = x
	elif x <= 1.25e+66:
		tmp = 0.0 - z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7e+74)
		tmp = x;
	elseif (x <= 1.25e+66)
		tmp = Float64(0.0 - z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7e+74)
		tmp = x;
	elseif (x <= 1.25e+66)
		tmp = 0.0 - z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7e+74], x, If[LessEqual[x, 1.25e+66], N[(0.0 - z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000029e74 or 1.24999999999999998e66 < x

   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 66.3%

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

   if -7.00000000000000029e74 < x < 1.24999999999999998e66

   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 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 41.9%

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

   5. Simplified 41.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 41.9%

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

   7. Applied egg-rr 41.9%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+66}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.0% 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 55.5%

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

Alternative 14: 30.4% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

5. Simplified 26.3%

   \[\leadsto \color{blue}{x} \]
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 2024158 
(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))