Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

Alternatives: 10

Speedup: N/A×

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

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

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(x - Float64(Float64(y + 0.5) * log(y))) + Float64(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[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+81}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \mathbf{elif}\;z \leq 180000000:\\ \;\;\;\;\left(x + y\right) - \left(y + 0.5\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.6e+81)
   (- (- y z) (* y (log y)))
   (if (<= z 180000000.0)
     (- (+ x y) (* (+ y 0.5) (log y)))
     (- (+ x (* (log y) -0.5)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.6e+81) {
		tmp = (y - z) - (y * log(y));
	} else if (z <= 180000000.0) {
		tmp = (x + y) - ((y + 0.5) * log(y));
	} else {
		tmp = (x + (log(y) * -0.5)) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.6d+81)) then
        tmp = (y - z) - (y * log(y))
    else if (z <= 180000000.0d0) then
        tmp = (x + y) - ((y + 0.5d0) * log(y))
    else
        tmp = (x + (log(y) * (-0.5d0))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.6e+81) {
		tmp = (y - z) - (y * Math.log(y));
	} else if (z <= 180000000.0) {
		tmp = (x + y) - ((y + 0.5) * Math.log(y));
	} else {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.6e+81:
		tmp = (y - z) - (y * math.log(y))
	elif z <= 180000000.0:
		tmp = (x + y) - ((y + 0.5) * math.log(y))
	else:
		tmp = (x + (math.log(y) * -0.5)) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.6e+81)
		tmp = Float64(Float64(y - z) - Float64(y * log(y)));
	elseif (z <= 180000000.0)
		tmp = Float64(Float64(x + y) - Float64(Float64(y + 0.5) * log(y)));
	else
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.6e+81)
		tmp = (y - z) - (y * log(y));
	elseif (z <= 180000000.0)
		tmp = (x + y) - ((y + 0.5) * log(y));
	else
		tmp = (x + (log(y) * -0.5)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.6e+81], N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 180000000.0], N[(N[(x + y), $MachinePrecision] - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.6e81

   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+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 84.7%

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

      1. log-rec 84.7%

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

   7. Simplified 84.7%

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

   if -6.6e81 < z < 1.8e8

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 98.4%

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

   if 1.8e8 < z

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 84.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+81}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \mathbf{elif}\;z \leq 180000000:\\ \;\;\;\;\left(x + y\right) - \left(y + 0.5\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+163} \lor \neg \left(z \leq 5.1 \cdot 10^{+15}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.12e+163) (not (<= z 5.1e+15)))
   (- x z)
   (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.12e+163) || !(z <= 5.1e+15)) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.12d+163)) .or. (.not. (z <= 5.1d+15))) then
        tmp = x - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.12e+163) || !(z <= 5.1e+15)) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.12e+163) or not (z <= 5.1e+15):
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.12e+163) || !(z <= 5.1e+15))
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.12e+163) || ~((z <= 5.1e+15)))
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.12e+163], N[Not[LessEqual[z, 5.1e+15]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.11999999999999996e163 or 5.1e15 < z

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 77.2%

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

      2. clear-num 77.2%

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

      3. sub-neg 77.2%

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

      4. metadata-eval 77.2%

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

      5. fma-neg 77.2%

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

      6. metadata-eval 77.2%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

      7. metadata-eval 77.2%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 77.2%

      \[\leadsto \left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + \left(y - z\right) \]
   7. Step-by-step derivation

      1. associate-+r- 77.2%

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y\right) + y\right) - z} \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 88.0%

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

   if -1.11999999999999996e163 < z < 5.1e15

   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 y around inf 98.1%

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

      1. +-commutative 98.1%

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

      2. associate--l+ 98.1%

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

      3. +-commutative 98.1%

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

      4. log-rec 98.1%

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

      5. unsub-neg 98.1%

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

      6. associate-*r/ 98.1%

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

      7. log-rec 98.1%

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

      8. distribute-rgt-neg-in 98.1%

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

      9. distribute-lft-neg-in 98.1%

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

      10. metadata-eval 98.1%

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

      11. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in y around inf 71.0%

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

      1. log-rec 71.0%

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

      2. mul-1-neg 71.0%

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

      3. remove-double-neg 71.0%

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

   10. Simplified 71.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+163} \lor \neg \left(z \leq 5.1 \cdot 10^{+15}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+22}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+129}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.2e+22)
   (- x z)
   (if (<= x 4.3e+129) (- (* y (- 1.0 (log y))) z) (- (+ x y) (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e+22) {
		tmp = x - z;
	} else if (x <= 4.3e+129) {
		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 <= (-4.2d+22)) then
        tmp = x - z
    else if (x <= 4.3d+129) 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 <= -4.2e+22) {
		tmp = x - z;
	} else if (x <= 4.3e+129) {
		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 <= -4.2e+22:
		tmp = x - z
	elif x <= 4.3e+129:
		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 <= -4.2e+22)
		tmp = Float64(x - z);
	elseif (x <= 4.3e+129)
		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 <= -4.2e+22)
		tmp = x - z;
	elseif (x <= 4.3e+129)
		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, -4.2e+22], N[(x - z), $MachinePrecision], If[LessEqual[x, 4.3e+129], 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 < -4.1999999999999996e22

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 81.0%

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

      2. clear-num 81.1%

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

      3. sub-neg 81.1%

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

      4. metadata-eval 81.1%

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

      5. fma-neg 81.1%

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

      6. metadata-eval 81.1%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

      7. metadata-eval 81.1%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 81.1%

      \[\leadsto \left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + \left(y - z\right) \]
   7. Step-by-step derivation

      1. associate-+r- 81.1%

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y\right) + y\right) - z} \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 84.9%

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

   if -4.1999999999999996e22 < x < 4.30000000000000021e129

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 71.3%

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

      1. log-rec 71.3%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in y around 0 71.2%

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

      1. neg-mul-1 71.2%

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

      2. sub-neg 71.2%

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

   10. Simplified 71.2%

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

   if 4.30000000000000021e129 < x

   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%

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

   3. Simplified 100.0%

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

      1. flip-+ 75.0%

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

      2. clear-num 75.0%

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

      3. sub-neg 75.0%

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

      4. metadata-eval 75.0%

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

      5. fma-neg 75.0%

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

      6. metadata-eval 75.0%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

      7. metadata-eval 75.0%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 75.0%

      \[\leadsto \left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + \left(y - z\right) \]

   7. Taylor expanded in y around inf 100.0%

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

   8. Taylor expanded in z around 0 99.9%

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

Alternative 5: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+22}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+129}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= x -1.45e+22) (- x z) (if (<= x 4.3e+129) (- t_0 z) (+ x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (x <= -1.45e+22) {
		tmp = x - z;
	} else if (x <= 4.3e+129) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (x <= (-1.45d+22)) then
        tmp = x - z
    else if (x <= 4.3d+129) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (x <= -1.45e+22) {
		tmp = x - z;
	} else if (x <= 4.3e+129) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if x <= -1.45e+22:
		tmp = x - z
	elif x <= 4.3e+129:
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (x <= -1.45e+22)
		tmp = Float64(x - z);
	elseif (x <= 4.3e+129)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (x <= -1.45e+22)
		tmp = x - z;
	elseif (x <= 4.3e+129)
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e+22], N[(x - z), $MachinePrecision], If[LessEqual[x, 4.3e+129], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e22

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 81.0%

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

      2. clear-num 81.1%

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

      3. sub-neg 81.1%

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

      4. metadata-eval 81.1%

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

      5. fma-neg 81.1%

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

      6. metadata-eval 81.1%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

      7. metadata-eval 81.1%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 81.1%

      \[\leadsto \left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + \left(y - z\right) \]
   7. Step-by-step derivation

      1. associate-+r- 81.1%

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y\right) + y\right) - z} \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 84.9%

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

   if -1.45e22 < x < 4.30000000000000021e129

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 71.3%

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

      1. log-rec 71.3%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in y around 0 71.2%

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

      1. neg-mul-1 71.2%

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

      2. sub-neg 71.2%

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

   10. Simplified 71.2%

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

   if 4.30000000000000021e129 < x

   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 inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. log-rec 100.0%

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

      5. unsub-neg 100.0%

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

      6. associate-*r/ 100.0%

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

      7. log-rec 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. metadata-eval 100.0%

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

      11. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 99.9%

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

      1. log-rec 99.9%

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

      2. mul-1-neg 99.9%

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

      3. remove-double-neg 99.9%

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

   10. Simplified 99.9%

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

Alternative 6: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 660000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 660000000.0)
   (- (+ x (* (log y) -0.5)) z)
   (- (+ x y) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 660000000.0) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = (x + y) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 660000000.0d0) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = (x + y) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 660000000.0) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 660000000.0:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 660000000.0)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - 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 (y <= 660000000.0)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 660000000.0], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6e8

   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 98.0%

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

   if 6.6e8 < y

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ 56.6%

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

      2. clear-num 56.5%

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

      3. sub-neg 56.5%

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

      4. metadata-eval 56.5%

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

      5. fma-neg 56.5%

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

      6. metadata-eval 56.5%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

      7. metadata-eval 56.5%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 56.5%

      \[\leadsto \left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + \left(y - z\right) \]

   7. Taylor expanded in y around inf 99.4%

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

   8. Taylor expanded in z around 0 78.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 660000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+176}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.3e+176) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e+176) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.3d+176) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e+176) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.3e+176:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.3e+176)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.3e+176)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.3e+176], 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 < 2.29999999999999996e176

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 94.9%

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

      2. clear-num 94.9%

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

      3. sub-neg 94.9%

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

      4. metadata-eval 94.9%

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

      5. fma-neg 94.9%

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

      6. metadata-eval 94.9%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

      7. metadata-eval 94.9%

         \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

   6. Applied egg-rr 94.9%

      \[\leadsto \left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + \left(y - z\right) \]
   7. Step-by-step derivation

      1. associate-+r- 94.9%

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y\right) + y\right) - z} \]

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around inf 65.1%

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

   if 2.29999999999999996e176 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.6%

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

      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 78.0%

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

      1. log-rec 78.0%

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

      2. sub-neg 78.0%

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

   7. Simplified 78.0%

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

Alternative 8: 48.2% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -72000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+129}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -72000000.0) x (if (<= x 4.3e+129) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -72000000.0) {
		tmp = x;
	} else if (x <= 4.3e+129) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-72000000.0d0)) then
        tmp = x
    else if (x <= 4.3d+129) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -72000000.0) {
		tmp = x;
	} else if (x <= 4.3e+129) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -72000000.0:
		tmp = x
	elif x <= 4.3e+129:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -72000000.0)
		tmp = x;
	elseif (x <= 4.3e+129)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -72000000.0)
		tmp = x;
	elseif (x <= 4.3e+129)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -72000000.0], x, If[LessEqual[x, 4.3e+129], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2e7 or 4.30000000000000021e129 < x

   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 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 70.8%

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

   if -7.2e7 < x < 4.30000000000000021e129

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 34.5%

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

      1. neg-mul-1 34.5%

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

   7. Simplified 34.5%

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

Alternative 9: 58.6% 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 99.9%

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

   1. associate--l+ 99.9%

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

3. Simplified 99.9%

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

   1. flip-+ 78.6%

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

   2. clear-num 78.6%

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

   3. sub-neg 78.6%

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

   4. metadata-eval 78.6%

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

   5. fma-neg 78.6%

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

   6. metadata-eval 78.6%

      \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

   7. metadata-eval 78.6%

      \[\leadsto \left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}} \cdot \log y\right) + \left(y - z\right) \]

6. Applied egg-rr 78.6%

   \[\leadsto \left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + \left(y - z\right) \]
7. Step-by-step derivation

   1. associate-+r- 78.6%

      \[\leadsto \color{blue}{\left(\left(x - \frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y\right) + y\right) - z} \]

8. Applied egg-rr 99.8%

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

9. Taylor expanded in x around inf 56.5%

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

Alternative 10: 30.7% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-+l+ 99.9%

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

   4. associate-+r- 99.9%

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

   5. *-commutative 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. fma-define 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 31.6%

   \[\leadsto \color{blue}{x} \]
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 2024138 
(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))