Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 12.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y \cdot \left(1 - \log y\right)\right) - \log y \cdot 0.5\right) - z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (- (+ x (* y (- 1.0 (log y)))) (* (log y) 0.5)) z))

C

double code(double x, double y, double z) {
	return ((x + (y * (1.0 - log(y)))) - (log(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 = ((x + (y * (1.0d0 - log(y)))) - (log(y) * 0.5d0)) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x + (y * (1.0 - Math.log(y)))) - (Math.log(y) * 0.5)) - z;
}

Python

def code(x, y, z):
	return ((x + (y * (1.0 - math.log(y)))) - (math.log(y) * 0.5)) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x + Float64(y * Float64(1.0 - log(y)))) - Float64(log(y) * 0.5)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x + (y * (1.0 - log(y)))) - (log(y) * 0.5)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * 0.5), $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. Taylor expanded in y around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \left(\left(x + y \cdot \left(1 - \log y\right)\right) - \log y \cdot 0.5\right) - z \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{+30}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.35e+30)
   (- (- x (* (log y) 0.5)) z)
   (+ x (* y (+ 1.0 (log (/ 1.0 y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.35e+30) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + (y * (1.0 + log((1.0 / 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 <= 2.35d+30) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + (y * (1.0d0 + log((1.0d0 / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.35e+30) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + (y * (1.0 + Math.log((1.0 / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.35e+30:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + (y * (1.0 + math.log((1.0 / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.35e+30)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 + log(Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.35e+30)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + (y * (1.0 + log((1.0 / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.35e+30], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[(1.0 + N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.34999999999999995e30

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   if 2.34999999999999995e30 < 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 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 99.7%

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

   6. Taylor expanded in y around inf 79.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{+30}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.000104:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.000104)
   (- (- 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 <= 0.000104) {
		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 <= 0.000104d0) 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 <= 0.000104) {
		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 <= 0.000104:
		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 <= 0.000104)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.000104)
		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, 0.000104], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.03999999999999994e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.8%

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

      1. *-commutative 98.8%

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

   5. Simplified 98.8%

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

   if 1.03999999999999994e-4 < 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 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. sub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.000104:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5400000:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5400000.0)
   (- (- x (* (log y) 0.5)) z)
   (- (- y (* (log y) (+ y 0.5))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5400000.0) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y - (log(y) * (y + 0.5))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5400000.0d0) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y - (log(y) * (y + 0.5d0))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5400000.0) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y - (Math.log(y) * (y + 0.5))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5400000.0:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y - (math.log(y) * (y + 0.5))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5400000.0)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y - Float64(log(y) * Float64(y + 0.5))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5400000.0)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y - (log(y) * (y + 0.5))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5400000.0], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.4e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.9%

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

      1. *-commutative 98.9%

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

   5. Simplified 98.9%

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

   if 5.4e6 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 79.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5400000:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 240000000:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 240000000.0) (- x z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 240000000.0) {
		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 (y <= 240000000.0d0) 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 (y <= 240000000.0) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 240000000.0:
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 240000000.0)
		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 (y <= 240000000.0)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 240000000.0], 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 y < 2.4e8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 71.9%

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

   if 2.4e8 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 80.0%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in y around inf 61.2%

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

      1. associate-*r/ 61.2%

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

      2. div-sub 61.2%

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

      3. mul-1-neg 61.2%

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

      4. log-rec 61.2%

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

      5. remove-double-neg 61.2%

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

   8. Simplified 61.2%

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

   9. Taylor expanded in x around 0 79.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 240000000:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.2e+82) (- (- x (* (log y) 0.5)) z) (- (- y (* y (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e+82) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y - (y * log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.2d+82) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y - (y * log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e+82) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y - (y * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.2e+82:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y - (y * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.2e+82)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y - Float64(y * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.2e+82)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y - (y * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.2e+82], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2e82

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 93.0%

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

      1. *-commutative 93.0%

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

   5. Simplified 93.0%

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

   if 4.2e82 < y

   1. Initial program 99.6%

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

      1. add-cube-cbrt 98.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right) \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}} - z \]

      2. pow3 98.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right)}^{3}} - z \]

      3. sub-neg 98.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y}\right)}^{3} - z \]

      4. associate-+l+ 98.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)}}\right)}^{3} - z \]

      5. *-commutative 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + y\right)}\right)}^{3} - z \]

      6. distribute-rgt-neg-in 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + y\right)}\right)}^{3} - z \]

      7. +-commutative 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right) + y\right)}\right)}^{3} - z \]

      8. distribute-neg-in 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} + y\right)}\right)}^{3} - z \]

      9. metadata-eval 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right) + y\right)}\right)}^{3} - z \]

      10. sub-neg 98.4%

          \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(-0.5 - y\right)} + y\right)}\right)}^{3} - z \]

      11. fma-undefine 98.5%

          \[\leadsto {\left(\sqrt[3]{x + \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)}}\right)}^{3} - z \]

   4. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)}\right)}^{3}} - z \]

   5. Taylor expanded in x around inf 76.9%

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

      1. +-commutative 76.9%

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

      2. mul-1-neg 76.9%

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

      3. unsub-neg 76.9%

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

      4. associate-/l* 76.9%

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

      5. +-commutative 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in y around inf 76.9%

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

      1. metadata-eval 76.9%

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

      2. times-frac 76.9%

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

      3. mul-1-neg 76.9%

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

      4. *-commutative 76.9%

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

      5. distribute-lft-neg-in 76.9%

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

      6. log-rec 76.9%

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

      7. remove-double-neg 76.9%

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

      8. times-frac 76.9%

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

      9. /-rgt-identity 76.9%

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

   10. Simplified 76.9%

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

   11. Taylor expanded in x around 0 84.2%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y \cdot \left(1 + \log \left(\frac{1}{y}\right)\right) - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (- (* y (+ 1.0 (log (/ 1.0 y)))) z)))

C

double code(double x, double y, double z) {
	return x + ((y * (1.0 + log((1.0 / 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 * (1.0d0 + log((1.0d0 / y)))) - z)
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * (1.0 + Math.log((1.0 / y)))) - z);
}

Python

def code(x, y, z):
	return x + ((y * (1.0 + math.log((1.0 / y)))) - z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(1.0 + log(Float64(1.0 / y)))) - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * (1.0 + log((1.0 / y)))) - z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(1.0 + N[Log[N[(1.0 / y), $MachinePrecision]], $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. 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 y around inf 84.8%

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

6. Final simplification 84.8%

   \[\leadsto x + \left(y \cdot \left(1 + \log \left(\frac{1}{y}\right)\right) - z\right) \]
7. 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: 71.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $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. Taylor expanded in y around 0 73.2%

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

   1. *-commutative 73.2%

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

5. Simplified 73.2%

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

6. Final simplification 73.2%

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

Alternative 11: 59.2% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf 58.3%

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

4. Final simplification 58.3%

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

Alternative 12: 30.8% accurate, 55.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 66.7%

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

4. Taylor expanded in z around inf 25.5%

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

   1. neg-mul-1 25.5%

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

6. Simplified 25.5%

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

7. Final simplification 25.5%

   \[\leadsto -z \]
8. Add Preprocessing

Developer target: 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 2024066 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))