Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.7s

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: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1060000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-127}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;x \leq 70000000000:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1060000000.0)
   (- x z)
   (if (<= x 6.4e-127)
     (- y (* (log y) (+ y 0.5)))
     (if (<= x 70000000000.0) (- (* (log y) -0.5) z) (- x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1060000000.0) {
		tmp = x - z;
	} else if (x <= 6.4e-127) {
		tmp = y - (log(y) * (y + 0.5));
	} else if (x <= 70000000000.0) {
		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 <= (-1060000000.0d0)) then
        tmp = x - z
    else if (x <= 6.4d-127) then
        tmp = y - (log(y) * (y + 0.5d0))
    else if (x <= 70000000000.0d0) 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 <= -1060000000.0) {
		tmp = x - z;
	} else if (x <= 6.4e-127) {
		tmp = y - (Math.log(y) * (y + 0.5));
	} else if (x <= 70000000000.0) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1060000000.0:
		tmp = x - z
	elif x <= 6.4e-127:
		tmp = y - (math.log(y) * (y + 0.5))
	elif x <= 70000000000.0:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1060000000.0)
		tmp = Float64(x - z);
	elseif (x <= 6.4e-127)
		tmp = Float64(y - Float64(log(y) * Float64(y + 0.5)));
	elseif (x <= 70000000000.0)
		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 <= -1060000000.0)
		tmp = x - z;
	elseif (x <= 6.4e-127)
		tmp = y - (log(y) * (y + 0.5));
	elseif (x <= 70000000000.0)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1060000000.0], N[(x - z), $MachinePrecision], If[LessEqual[x, 6.4e-127], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 70000000000.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.06e9 or 7e10 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. log-rec 99.8%

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

      4. remove-double-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 84.3%

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

   if -1.06e9 < x < 6.40000000000000035e-127

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

      2. neg-mul-1 98.8%

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

      3. +-commutative 98.8%

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

      4. cancel-sign-sub-inv 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in z around 0 69.6%

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

      1. +-commutative 69.6%

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

   10. Simplified 69.6%

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

   if 6.40000000000000035e-127 < x < 7e10

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. neg-mul-1 98.4%

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

      3. +-commutative 98.4%

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

      4. cancel-sign-sub-inv 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in y around 0 71.0%

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

      1. *-commutative 71.0%

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

   10. Simplified 71.0%

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

Alternative 3: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+19} \lor \neg \left(x \leq 7.5 \cdot 10^{+55}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.25e+19) (not (<= x 7.5e+55)))
   (- x z)
   (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e+19) || !(x <= 7.5e+55)) {
		tmp = x - 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 ((x <= (-1.25d+19)) .or. (.not. (x <= 7.5d+55))) then
        tmp = x - 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 ((x <= -1.25e+19) || !(x <= 7.5e+55)) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.25e+19) or not (x <= 7.5e+55):
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.25e+19) || !(x <= 7.5e+55))
		tmp = Float64(x - 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 ((x <= -1.25e+19) || ~((x <= 7.5e+55)))
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.25e+19], N[Not[LessEqual[x, 7.5e+55]], $MachinePrecision]], N[(x - 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 x < -1.25e19 or 7.50000000000000014e55 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. log-rec 99.8%

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

      4. remove-double-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 86.3%

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

   if -1.25e19 < x < 7.50000000000000014e55

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. distribute-rgt-neg-in 75.6%

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

      3. log-rec 75.6%

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

      4. remove-double-neg 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in x around 0 74.2%

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

   9. Taylor expanded in y around 0 74.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+19} \lor \neg \left(x \leq 7.5 \cdot 10^{+55}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+19} \lor \neg \left(x \leq 2.25 \cdot 10^{+57}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3e+19) (not (<= x 2.25e+57)))
   (- x z)
   (- y (+ z (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e+19) || !(x <= 2.25e+57)) {
		tmp = x - z;
	} else {
		tmp = y - (z + (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3d+19)) .or. (.not. (x <= 2.25d+57))) then
        tmp = x - z
    else
        tmp = y - (z + (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3e+19) || !(x <= 2.25e+57)) {
		tmp = x - z;
	} else {
		tmp = y - (z + (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3e+19) or not (x <= 2.25e+57):
		tmp = x - z
	else:
		tmp = y - (z + (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3e+19) || !(x <= 2.25e+57))
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(z + Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3e+19) || ~((x <= 2.25e+57)))
		tmp = x - z;
	else
		tmp = y - (z + (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3e+19], N[Not[LessEqual[x, 2.25e+57]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(y - N[(z + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3e19 or 2.24999999999999998e57 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. log-rec 99.8%

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

      4. remove-double-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 86.3%

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

   if -3e19 < x < 2.24999999999999998e57

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. distribute-rgt-neg-in 75.6%

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

      3. log-rec 75.6%

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

      4. remove-double-neg 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in x around 0 74.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+19} \lor \neg \left(x \leq 2.25 \cdot 10^{+57}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;y \leq 6.6 \cdot 10^{+71}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+175}:\\ \;\;\;\;y - \left(z + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= y 6.6e+71)
     (- (+ x (* (log y) -0.5)) z)
     (if (<= y 3.3e+175) (- y (+ z t_0)) (- (+ x y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (y <= 6.6e+71) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if (y <= 3.3e+175) {
		tmp = y - (z + t_0);
	} else {
		tmp = (x + y) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * log(y)
    if (y <= 6.6d+71) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if (y <= 3.3d+175) then
        tmp = y - (z + t_0)
    else
        tmp = (x + y) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log(y);
	double tmp;
	if (y <= 6.6e+71) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if (y <= 3.3e+175) {
		tmp = y - (z + t_0);
	} else {
		tmp = (x + y) - t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if y <= 6.6e+71:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif y <= 3.3e+175:
		tmp = y - (z + t_0)
	else:
		tmp = (x + y) - t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (y <= 6.6e+71)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif (y <= 3.3e+175)
		tmp = Float64(y - Float64(z + t_0));
	else
		tmp = Float64(Float64(x + y) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (y <= 6.6e+71)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif (y <= 3.3e+175)
		tmp = y - (z + t_0);
	else
		tmp = (x + y) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.6e+71], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 3.3e+175], N[(y - N[(z + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.5999999999999996e71

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 93.0%

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

   if 6.5999999999999996e71 < y < 3.3000000000000002e175

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. log-rec 99.7%

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

      4. remove-double-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 89.0%

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

   if 3.3000000000000002e175 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. distribute-rgt-neg-in 99.4%

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

      3. log-rec 99.4%

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

      4. remove-double-neg 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in z around 0 94.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+71}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+175}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+19}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+54}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.25e+19)
   (- x z)
   (if (<= x 7.8e+54) (- (* y (- 1.0 (log y))) z) (- (+ x y) (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e+19) {
		tmp = x - z;
	} else if (x <= 7.8e+54) {
		tmp = (y * (1.0 - log(y))) - z;
	} else {
		tmp = (x + y) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.25d+19)) then
        tmp = x - z
    else if (x <= 7.8d+54) then
        tmp = (y * (1.0d0 - log(y))) - z
    else
        tmp = (x + y) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e+19) {
		tmp = x - z;
	} else if (x <= 7.8e+54) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.25e+19:
		tmp = x - z
	elif x <= 7.8e+54:
		tmp = (y * (1.0 - math.log(y))) - z
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.25e+19)
		tmp = Float64(x - z);
	elseif (x <= 7.8e+54)
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	else
		tmp = Float64(Float64(x + y) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.25e+19)
		tmp = x - z;
	elseif (x <= 7.8e+54)
		tmp = (y * (1.0 - log(y))) - z;
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.25e+19], N[(x - z), $MachinePrecision], If[LessEqual[x, 7.8e+54], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e19

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. log-rec 99.9%

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

      4. remove-double-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 91.4%

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

   if -1.25e19 < x < 7.8000000000000005e54

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. distribute-rgt-neg-in 75.6%

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

      3. log-rec 75.6%

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

      4. remove-double-neg 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in x around 0 74.2%

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

   9. Taylor expanded in y around 0 74.2%

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

   if 7.8000000000000005e54 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. log-rec 99.7%

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

      4. remove-double-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around 0 86.1%

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

Alternative 7: 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. Step-by-step derivation

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.6%

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

   if 0.28000000000000003 < 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. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. distribute-rgt-neg-in 98.0%

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

      3. log-rec 98.0%

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

      4. remove-double-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in y around 0 98.0%

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

4. Final simplification 98.8%

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

Alternative 8: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.2:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \left(\left(x - y \cdot \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.2) (- (+ 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 <= 0.2) {
		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 <= 0.2d0) 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 <= 0.2) {
		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 <= 0.2:
		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 <= 0.2)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(y + Float64(Float64(x - Float64(y * log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.2)
		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, 0.2], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y + N[(N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.20000000000000001

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.6%

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

   if 0.20000000000000001 < 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. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. distribute-rgt-neg-in 98.0%

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

      3. log-rec 98.0%

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

      4. remove-double-neg 98.0%

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

   7. Simplified 98.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.2:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \left(\left(x - y \cdot \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. associate--l+ 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;y + \left(x - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e+87) (+ y (- x z)) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+87) {
		tmp = y + (x - z);
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.5d+87) then
        tmp = y + (x - z)
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+87) {
		tmp = y + (x - z);
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e+87:
		tmp = y + (x - z)
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e+87)
		tmp = Float64(y + Float64(x - z));
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.5e+87)
		tmp = y + (x - z);
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e+87], N[(y + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5000000000000003e87

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. distribute-rgt-neg-in 78.9%

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

      3. log-rec 78.9%

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

      4. remove-double-neg 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in x around inf 72.7%

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

   if 4.5000000000000003e87 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-+r- 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-define 99.6%

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

      8. +-commutative 99.6%

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

      9. distribute-neg-in 99.6%

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

      10. unsub-neg 99.6%

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 66.8%

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

      1. log-rec 66.8%

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

      2. sub-neg 66.8%

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

   7. Simplified 66.8%

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

Alternative 12: 47.7% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+163} \lor \neg \left(z \leq 220\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.2e+163) (not (<= z 220.0))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.2e+163) || !(z <= 220.0)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.2d+163)) .or. (.not. (z <= 220.0d0))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.2e+163) || !(z <= 220.0)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.2e+163) or not (z <= 220.0):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.2e+163) || !(z <= 220.0))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.2e+163) || ~((z <= 220.0)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.2e+163], N[Not[LessEqual[z, 220.0]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -1.1999999999999999e163 or 220 < z

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 68.1%

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

      1. neg-mul-1 68.1%

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

   7. Simplified 68.1%

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

   if -1.1999999999999999e163 < z < 220

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 41.9%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+163} \lor \neg \left(z \leq 220\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.1% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. associate--l+ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around inf 86.3%

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

   1. mul-1-neg 86.3%

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

   2. distribute-rgt-neg-in 86.3%

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

   3. log-rec 86.3%

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

   4. remove-double-neg 86.3%

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

7. Simplified 86.3%

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

8. Taylor expanded in y around 0 59.2%

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

Alternative 14: 30.5% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 32.3%

   \[\leadsto \color{blue}{x} \]
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 2024180 
(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))