Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 15.2s

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(y + x), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(1.0 / N[(-1.0 / N[(y + 0.5), $MachinePrecision]), $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. Step-by-step derivation

   1. +-commutative 99.8%

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

   2. associate-+r- 99.8%

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

   3. *-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. distribute-rgt-in 99.8%

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

   6. associate--r+ 99.8%

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

   7. *-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+107} \lor \neg \left(y \leq 4.5 \cdot 10^{+161}\right):\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 6.5e+42)
     (- (+ y x) z)
     (if (or (<= y 2.7e+107) (not (<= y 4.5e+161))) (- t_0 z) (+ x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 6.5e+42) {
		tmp = (y + x) - z;
	} else if ((y <= 2.7e+107) || !(y <= 4.5e+161)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 6.5d+42) then
        tmp = (y + x) - z
    else if ((y <= 2.7d+107) .or. (.not. (y <= 4.5d+161))) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 6.5e+42) {
		tmp = (y + x) - z;
	} else if ((y <= 2.7e+107) || !(y <= 4.5e+161)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 6.5e+42:
		tmp = (y + x) - z
	elif (y <= 2.7e+107) or not (y <= 4.5e+161):
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 6.5e+42)
		tmp = Float64(Float64(y + x) - z);
	elseif ((y <= 2.7e+107) || !(y <= 4.5e+161))
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 6.5e+42)
		tmp = (y + x) - z;
	elseif ((y <= 2.7e+107) || ~((y <= 4.5e+161)))
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.5e+42], N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 2.7e+107], N[Not[LessEqual[y, 4.5e+161]], $MachinePrecision]], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.50000000000000052e42

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-+ 99.9%

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

      3. associate-*r/ 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf 79.0%

      \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{y} - 0.5 \cdot \frac{1}{{y}^{2}}}}\right) + y\right) - z \]
   7. Step-by-step derivation

      1. associate-*r/ 79.0%

         \[\leadsto \left(\left(x - \frac{\log y}{\frac{1}{y} - \color{blue}{\frac{0.5 \cdot 1}{{y}^{2}}}}\right) + y\right) - z \]

      2. metadata-eval 79.0%

         \[\leadsto \left(\left(x - \frac{\log y}{\frac{1}{y} - \frac{\color{blue}{0.5}}{{y}^{2}}}\right) + y\right) - z \]

      3. unpow2 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in y around 0 75.7%

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

       1. +-commutative 75.7%

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

   11. Simplified 75.7%

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

   if 6.50000000000000052e42 < y < 2.7000000000000001e107 or 4.49999999999999992e161 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r- 99.6%

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

      3. *-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-rgt-in 99.6%

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

      6. associate--r+ 99.6%

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

      7. *-commutative 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around inf 91.0%

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

      1. log-rec 91.0%

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

      2. mul-1-neg 91.0%

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

      3. remove-double-neg 91.0%

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

   6. Simplified 91.0%

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

   if 2.7000000000000001e107 < y < 4.49999999999999992e161

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. *-commutative 99.4%

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

      8. fma-def 99.5%

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

      9. neg-sub0 99.5%

         \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

      10. +-commutative 99.5%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

      11. associate--r+ 99.5%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

      12. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in z around 0 85.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+107} \lor \neg \left(y \leq 4.5 \cdot 10^{+161}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+107} \lor \neg \left(y \leq 3.4 \cdot 10^{+161}\right):\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 8e+42)
     (- (- x (* (log y) 0.5)) z)
     (if (or (<= y 3.3e+107) (not (<= y 3.4e+161))) (- t_0 z) (+ x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 8e+42) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if ((y <= 3.3e+107) || !(y <= 3.4e+161)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 8d+42) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else if ((y <= 3.3d+107) .or. (.not. (y <= 3.4d+161))) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 8e+42) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if ((y <= 3.3e+107) || !(y <= 3.4e+161)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 8e+42:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif (y <= 3.3e+107) or not (y <= 3.4e+161):
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 8e+42)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif ((y <= 3.3e+107) || !(y <= 3.4e+161))
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 8e+42)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif ((y <= 3.3e+107) || ~((y <= 3.4e+161)))
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 8e+42], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 3.3e+107], N[Not[LessEqual[y, 3.4e+161]], $MachinePrecision]], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.00000000000000036e42

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 96.0%

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

   if 8.00000000000000036e42 < y < 3.30000000000000032e107 or 3.39999999999999993e161 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r- 99.6%

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

      3. *-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-rgt-in 99.6%

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

      6. associate--r+ 99.6%

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

      7. *-commutative 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in y around inf 91.0%

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

      1. log-rec 91.0%

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

      2. mul-1-neg 91.0%

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

      3. remove-double-neg 91.0%

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

   6. Simplified 91.0%

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

   if 3.30000000000000032e107 < y < 3.39999999999999993e161

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. *-commutative 99.4%

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

      8. fma-def 99.5%

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

      9. neg-sub0 99.5%

         \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

      10. +-commutative 99.5%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

      11. associate--r+ 99.5%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

      12. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in z around 0 85.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+42}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+107} \lor \neg \left(y \leq 3.4 \cdot 10^{+161}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 4: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 4.7 \cdot 10^{+42}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+107}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+162}:\\ \;\;\;\;x + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 4.7e+42)
     (- (- x (* (log y) 0.5)) z)
     (if (<= y 2.15e+107)
       (- (- y (* y (log y))) z)
       (if (<= y 8e+162) (+ x t_0) (- t_0 z))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 4.7e+42) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if (y <= 2.15e+107) {
		tmp = (y - (y * log(y))) - z;
	} else if (y <= 8e+162) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 4.7d+42) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else if (y <= 2.15d+107) then
        tmp = (y - (y * log(y))) - z
    else if (y <= 8d+162) then
        tmp = x + t_0
    else
        tmp = t_0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 4.7e+42) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if (y <= 2.15e+107) {
		tmp = (y - (y * Math.log(y))) - z;
	} else if (y <= 8e+162) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 4.7e+42:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif y <= 2.15e+107:
		tmp = (y - (y * math.log(y))) - z
	elif y <= 8e+162:
		tmp = x + t_0
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 4.7e+42)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif (y <= 2.15e+107)
		tmp = Float64(Float64(y - Float64(y * log(y))) - z);
	elseif (y <= 8e+162)
		tmp = Float64(x + t_0);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 4.7e+42)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif (y <= 2.15e+107)
		tmp = (y - (y * log(y))) - z;
	elseif (y <= 8e+162)
		tmp = x + t_0;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.7e+42], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 2.15e+107], N[(N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 8e+162], N[(x + t$95$0), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.69999999999999986e42

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 96.0%

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

   if 4.69999999999999986e42 < y < 2.15e107

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around inf 84.2%

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

      1. log-rec 84.2%

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

   4. Simplified 84.2%

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

   if 2.15e107 < y < 7.9999999999999995e162

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-neg-in 99.4%

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

      7. *-commutative 99.4%

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

      8. fma-def 99.5%

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

      9. neg-sub0 99.5%

         \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

      10. +-commutative 99.5%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

      11. associate--r+ 99.5%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

      12. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in z around 0 85.6%

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

   if 7.9999999999999995e162 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-+r- 99.5%

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

      3. *-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-rgt-in 99.5%

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

      6. associate--r+ 99.5%

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

      7. *-commutative 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around inf 93.3%

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

      1. log-rec 93.3%

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

      2. mul-1-neg 93.3%

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

      3. remove-double-neg 93.3%

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

   6. Simplified 93.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{+42}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+107}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+162}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.063:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.063)
   (- (- x (* (log y) 0.5)) z)
   (+ (- x z) (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.063) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (x - z) + (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 <= 0.063d0) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (x - z) + (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 <= 0.063) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (x - z) + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.063:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (x - z) + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.063)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(x - z) + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.063)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (x - z) + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.063], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x - z), $MachinePrecision] + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.063

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 99.1%

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

   if 0.063 < 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. sub-neg 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-lft-neg-in 99.6%

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

      7. *-commutative 99.6%

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

      8. fma-def 99.6%

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

      9. neg-sub0 99.6%

         \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

      10. +-commutative 99.6%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

      11. associate--r+ 99.6%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. sub-neg 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.063:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 6: 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. Final simplification 99.8%

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

Alternative 7: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -160 \lor \neg \left(x \leq 270000000000\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -160.0) (not (<= x 270000000000.0)))
   (- x z)
   (- (* (log y) -0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -160.0) || !(x <= 270000000000.0)) {
		tmp = x - z;
	} else {
		tmp = (log(y) * -0.5) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-160.0d0)) .or. (.not. (x <= 270000000000.0d0))) then
        tmp = x - z
    else
        tmp = (log(y) * (-0.5d0)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -160.0) || !(x <= 270000000000.0)) {
		tmp = x - z;
	} else {
		tmp = (Math.log(y) * -0.5) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -160.0) or not (x <= 270000000000.0):
		tmp = x - z
	else:
		tmp = (math.log(y) * -0.5) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -160.0) || !(x <= 270000000000.0))
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(log(y) * -0.5) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -160.0) || ~((x <= 270000000000.0)))
		tmp = x - z;
	else
		tmp = (log(y) * -0.5) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -160.0], N[Not[LessEqual[x, 270000000000.0]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -160 or 2.7e11 < 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. +-commutative 99.9%

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

      2. associate-+r- 99.9%

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

      3. *-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. associate--r+ 99.9%

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

      7. *-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 75.7%

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

   if -160 < x < 2.7e11

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-+r- 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-rgt-in 99.7%

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

      6. associate--r+ 99.7%

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

      7. *-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. distribute-rgt-in 99.1%

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

      4. distribute-rgt-neg-out 99.1%

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

      5. +-commutative 99.1%

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

      6. fma-def 99.1%

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

      7. distribute-neg-in 99.1%

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

      8. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in y around 0 62.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -160 \lor \neg \left(x \leq 270000000000\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]

Alternative 8: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -50000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 165:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -50000000000.0)
   (- x z)
   (if (<= z 165.0) (- x (* (log y) 0.5)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -50000000000.0) {
		tmp = x - z;
	} else if (z <= 165.0) {
		tmp = x - (log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-50000000000.0d0)) then
        tmp = x - z
    else if (z <= 165.0d0) then
        tmp = x - (log(y) * 0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -50000000000.0) {
		tmp = x - z;
	} else if (z <= 165.0) {
		tmp = x - (Math.log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -50000000000.0:
		tmp = x - z
	elif z <= 165.0:
		tmp = x - (math.log(y) * 0.5)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -50000000000.0)
		tmp = Float64(x - z);
	elseif (z <= 165.0)
		tmp = Float64(x - Float64(log(y) * 0.5));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -50000000000.0)
		tmp = x - z;
	elseif (z <= 165.0)
		tmp = x - (log(y) * 0.5);
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -50000000000.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 165.0], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5e10 or 165 < z

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+r- 99.8%

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

      3. *-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. associate--r+ 99.8%

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

      7. *-commutative 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 76.3%

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

   if -5e10 < z < 165

   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. sub-neg 99.6%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-lft-neg-in 99.6%

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

      7. *-commutative 99.6%

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

      8. fma-def 99.7%

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

      9. neg-sub0 99.7%

         \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

      10. +-commutative 99.7%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

      11. associate--r+ 99.7%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

      12. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. sub-neg 98.6%

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

      3. associate--l+ 98.6%

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

      4. cancel-sign-sub-inv 98.6%

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

      5. +-commutative 98.6%

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

      6. +-commutative 98.6%

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

      7. cancel-sign-sub-inv 98.6%

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

      8. associate--l+ 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in y around 0 57.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -50000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 165:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 9: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+70}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.8e+70) (- (+ y x) z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e+70) {
		tmp = (y + x) - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.8d+70) then
        tmp = (y + x) - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e+70) {
		tmp = (y + x) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.8e+70:
		tmp = (y + x) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.8e+70)
		tmp = Float64(Float64(y + x) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.8e+70)
		tmp = (y + x) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.8e+70], N[(N[(y + x), $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 < 6.8000000000000002e70

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-+ 99.9%

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

      3. associate-*r/ 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-/l* 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf 81.3%

      \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{y} - 0.5 \cdot \frac{1}{{y}^{2}}}}\right) + y\right) - z \]
   7. Step-by-step derivation

      1. associate-*r/ 81.3%

         \[\leadsto \left(\left(x - \frac{\log y}{\frac{1}{y} - \color{blue}{\frac{0.5 \cdot 1}{{y}^{2}}}}\right) + y\right) - z \]

      2. metadata-eval 81.3%

         \[\leadsto \left(\left(x - \frac{\log y}{\frac{1}{y} - \frac{\color{blue}{0.5}}{{y}^{2}}}\right) + y\right) - z \]

      3. unpow2 81.3%

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

   8. Simplified 81.3%

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

   9. Taylor expanded in y around 0 74.3%

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

       1. +-commutative 74.3%

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

   11. Simplified 74.3%

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

   if 6.8000000000000002e70 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-lft-neg-in 99.5%

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

      7. *-commutative 99.5%

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

      8. fma-def 99.6%

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

      9. neg-sub0 99.6%

         \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

      10. +-commutative 99.6%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

      11. associate--r+ 99.6%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

      12. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in z around 0 81.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+70}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 10: 46.0% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+175}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e+30) x (if (<= x 1.75e+175) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+30) {
		tmp = x;
	} else if (x <= 1.75e+175) {
		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 <= (-1d+30)) then
        tmp = x
    else if (x <= 1.75d+175) 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 <= -1e+30) {
		tmp = x;
	} else if (x <= 1.75e+175) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e+30:
		tmp = x
	elif x <= 1.75e+175:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e+30)
		tmp = x;
	elseif (x <= 1.75e+175)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e+30)
		tmp = x;
	elseif (x <= 1.75e+175)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e+30], x, If[LessEqual[x, 1.75e+175], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1e30 or 1.7500000000000002e175 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-neg-in 99.8%

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

      7. *-commutative 99.8%

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

      8. fma-def 99.9%

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

      9. neg-sub0 99.9%

         \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

      10. +-commutative 99.9%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

      11. associate--r+ 99.9%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

      12. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 64.5%

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

   if -1e30 < x < 1.7500000000000002e175

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-lft-neg-in 99.7%

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

      7. *-commutative 99.7%

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

      8. fma-def 99.7%

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

      9. neg-sub0 99.7%

         \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

      10. +-commutative 99.7%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

      11. associate--r+ 99.7%

          \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

      12. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around inf 43.0%

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

      1. mul-1-neg 43.0%

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

   6. Simplified 43.0%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+30}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+175}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 58.2% 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. Step-by-step derivation

   1. +-commutative 99.8%

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

   2. associate-+r- 99.8%

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

   3. *-commutative 99.8%

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

   4. +-commutative 99.8%

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

   5. distribute-rgt-in 99.8%

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

   6. associate--r+ 99.8%

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

   7. *-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in x around inf 57.8%

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

5. Final simplification 57.8%

   \[\leadsto x - z \]

Alternative 12: 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. sub-neg 99.8%

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

   2. associate-+l+ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-lft-neg-in 99.8%

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

   7. *-commutative 99.8%

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

   8. fma-def 99.8%

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

   9. neg-sub0 99.8%

      \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, y\right) \]

   10. +-commutative 99.8%

       \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, y\right) \]

   11. associate--r+ 99.8%

       \[\leadsto \left(x - z\right) + \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, y\right) \]

   12. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 26.5%

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

5. Final simplification 26.5%

   \[\leadsto x \]

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 2023297 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

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