Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+82}:\\ \;\;\;\;x + z \cdot \left(-0.5 \cdot \frac{\log y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \log \left(\frac{e}{y}\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.6e+82)
   (+ x (* z (+ (* -0.5 (/ (log y) z)) -1.0)))
   (- (* y (log (/ E y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.6e+82) {
		tmp = x + (z * ((-0.5 * (log(y) / z)) + -1.0));
	} else {
		tmp = (y * log((((double) M_E) / y))) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.6e+82) {
		tmp = x + (z * ((-0.5 * (Math.log(y) / z)) + -1.0));
	} else {
		tmp = (y * Math.log((Math.E / y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.6e+82:
		tmp = x + (z * ((-0.5 * (math.log(y) / z)) + -1.0))
	else:
		tmp = (y * math.log((math.e / y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.6e+82)
		tmp = Float64(x + Float64(z * Float64(Float64(-0.5 * Float64(log(y) / z)) + -1.0)));
	else
		tmp = Float64(Float64(y * log(Float64(exp(1) / y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.6e+82)
		tmp = x + (z * ((-0.5 * (log(y) / z)) + -1.0));
	else
		tmp = (y * log((2.71828182845904523536 / y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.6e+82], N[(x + N[(z * N[(N[(-0.5 * N[(N[Log[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.6000000000000001e82

   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 z around inf 98.7%

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

      1. associate-+r- 98.7%

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

      2. associate--l+ 98.7%

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

      3. sub-neg 98.7%

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

      4. fma-define 98.7%

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

      5. associate-/l* 98.5%

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

      6. +-commutative 98.5%

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

      7. metadata-eval 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in y around 0 93.1%

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

   if 5.6000000000000001e82 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 88.6%

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

      1. log-rec 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around 0 88.6%

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

      1. neg-mul-1 88.6%

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

      2. sub-neg 88.6%

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

   8. Simplified 88.6%

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

      1. add-log-exp 88.6%

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

      2. exp-diff 88.6%

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

      3. add-exp-log 88.6%

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

   10. Applied egg-rr 88.6%

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

       1. exp-1-e 88.6%

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

   12. Simplified 88.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+82}:\\ \;\;\;\;x + z \cdot \left(-0.5 \cdot \frac{\log y}{z} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \log \left(\frac{e}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+80}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \log \left(\frac{e}{y}\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e+80) (- (- x (* 0.5 (log y))) z) (- (* y (log (/ E y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+80) {
		tmp = (x - (0.5 * log(y))) - z;
	} else {
		tmp = (y * log((((double) M_E) / y))) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+80) {
		tmp = (x - (0.5 * Math.log(y))) - z;
	} else {
		tmp = (y * Math.log((Math.E / y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.2e+80:
		tmp = (x - (0.5 * math.log(y))) - z
	else:
		tmp = (y * math.log((math.e / y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e+80)
		tmp = Float64(Float64(x - Float64(0.5 * log(y))) - z);
	else
		tmp = Float64(Float64(y * log(Float64(exp(1) / y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.2e+80)
		tmp = (x - (0.5 * log(y))) - z;
	else
		tmp = (y * log((2.71828182845904523536 / y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e+80], N[(N[(x - N[(0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.20000000000000016e80

   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 93.1%

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

   if 9.20000000000000016e80 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 88.6%

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

      1. log-rec 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around 0 88.6%

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

      1. neg-mul-1 88.6%

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

      2. sub-neg 88.6%

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

   8. Simplified 88.6%

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

      1. add-log-exp 88.6%

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

      2. exp-diff 88.6%

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

      3. add-exp-log 88.6%

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

   10. Applied egg-rr 88.6%

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

       1. exp-1-e 88.6%

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

   12. Simplified 88.6%

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

Alternative 4: 76.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+81) (- (+ x y) z) (- (* y (log (/ E y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+81) {
		tmp = (x + y) - z;
	} else {
		tmp = (y * log((((double) M_E) / y))) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+81) {
		tmp = (x + y) - z;
	} else {
		tmp = (y * Math.log((Math.E / y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1e+81:
		tmp = (x + y) - z
	else:
		tmp = (y * math.log((math.e / y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+81)
		tmp = Float64(Float64(x + y) - z);
	else
		tmp = Float64(Float64(y * log(Float64(exp(1) / y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e+81)
		tmp = (x + y) - z;
	else
		tmp = (y * log((2.71828182845904523536 / y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+81], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999921e80

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. sub-neg 98.8%

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

      3. associate-/l* 98.8%

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

      4. +-commutative 98.8%

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

      5. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf 71.8%

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

      1. neg-mul-1 71.8%

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

   8. Simplified 71.8%

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

   if 9.99999999999999921e80 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 88.6%

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

      1. log-rec 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around 0 88.6%

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

      1. neg-mul-1 88.6%

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

      2. sub-neg 88.6%

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

   8. Simplified 88.6%

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

      1. add-log-exp 88.6%

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

      2. exp-diff 88.6%

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

      3. add-exp-log 88.6%

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

   10. Applied egg-rr 88.6%

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

       1. exp-1-e 88.6%

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

   12. Simplified 88.6%

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

4. Final simplification 77.9%

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

Alternative 5: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+94}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.6e+94) (- (+ x y) z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.6e+94) {
		tmp = (x + y) - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.6d+94) then
        tmp = (x + y) - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.6e+94) {
		tmp = (x + y) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.6e+94:
		tmp = (x + y) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.6e+94)
		tmp = Float64(Float64(x + y) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.6e+94)
		tmp = (x + y) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.6e+94], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999999e94

   1. Initial program 99.9%

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

   3. Taylor expanded in x around -inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. sub-neg 98.8%

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

      3. associate-/l* 98.8%

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

      4. +-commutative 98.8%

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

      5. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf 72.1%

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

      1. neg-mul-1 72.1%

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

   8. Simplified 72.1%

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

   if 4.5999999999999999e94 < 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 z around inf 71.3%

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

      1. associate-+r- 71.2%

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

      2. associate--l+ 71.3%

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

      3. sub-neg 71.3%

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

      4. fma-define 71.3%

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

      5. associate-/l* 71.1%

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

      6. +-commutative 71.1%

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

      7. metadata-eval 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in y around inf 57.7%

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

      1. log-rec 57.7%

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

      2. distribute-frac-neg 57.7%

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

      3. unsub-neg 57.7%

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

   10. Simplified 57.7%

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

   11. Taylor expanded in z around -inf 84.2%

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

   12. Taylor expanded in y around inf 84.2%

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

       1. log-rec 84.2%

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

       2. sub-neg 84.2%

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

   14. Simplified 84.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+94}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{+105}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.08e+105) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.08e+105) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.08d+105) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.08e+105) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.08e+105:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.08e+105)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.08e+105)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.08e+105], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.07999999999999994e105

   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 z around inf 98.2%

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

      1. associate-+r- 98.2%

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

      2. associate--l+ 98.2%

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

      3. sub-neg 98.2%

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

      4. fma-define 98.2%

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

      5. associate-/l* 98.0%

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

      6. +-commutative 98.0%

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

      7. metadata-eval 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in z around inf 71.5%

      \[\leadsto x + z \cdot \color{blue}{-1} \]

   if 1.07999999999999994e105 < 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 75.3%

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

      1. log-rec 75.3%

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

      2. sub-neg 75.3%

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

   7. Simplified 75.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.08 \cdot 10^{+105}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.8% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+71}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.75e+66) x (if (<= x 5.8e+71) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.75e+66) {
		tmp = x;
	} else if (x <= 5.8e+71) {
		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 <= (-2.75d+66)) then
        tmp = x
    else if (x <= 5.8d+71) 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 <= -2.75e+66) {
		tmp = x;
	} else if (x <= 5.8e+71) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.75e+66:
		tmp = x
	elif x <= 5.8e+71:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.75e+66)
		tmp = x;
	elseif (x <= 5.8e+71)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.75e+66)
		tmp = x;
	elseif (x <= 5.8e+71)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.75e+66], x, If[LessEqual[x, 5.8e+71], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.75e66 or 5.80000000000000014e71 < x

   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 63.4%

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

   if -2.75e66 < x < 5.80000000000000014e71

   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 39.6%

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

      1. neg-mul-1 39.6%

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

   7. Simplified 39.6%

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

Alternative 8: 57.1% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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 89.3%

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

   1. associate-+r- 89.3%

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

   2. associate--l+ 89.3%

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

   3. sub-neg 89.3%

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

   4. fma-define 89.3%

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

   5. associate-/l* 89.1%

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

   6. +-commutative 89.1%

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

   7. metadata-eval 89.1%

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

7. Simplified 89.1%

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

8. Taylor expanded in z around inf 56.1%

   \[\leadsto x + z \cdot \color{blue}{-1} \]

9. Final simplification 56.1%

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

Alternative 9: 29.5% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 27.0%

   \[\leadsto \color{blue}{x} \]
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 2024170 
(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))