Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+148} \lor \neg \left(y \leq 2.6 \cdot 10^{+163}\right) \land y \leq 1.3 \cdot 10^{+202}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 3.3e+148) (and (not (<= y 2.6e+163)) (<= y 1.3e+202)))
   (- x z)
   (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 3.3e+148) || (!(y <= 2.6e+163) && (y <= 1.3e+202))) {
		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 <= 3.3d+148) .or. (.not. (y <= 2.6d+163)) .and. (y <= 1.3d+202)) 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 <= 3.3e+148) || (!(y <= 2.6e+163) && (y <= 1.3e+202))) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 3.3e+148) or (not (y <= 2.6e+163) and (y <= 1.3e+202)):
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 3.3e+148) || (!(y <= 2.6e+163) && (y <= 1.3e+202)))
		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 <= 3.3e+148) || (~((y <= 2.6e+163)) && (y <= 1.3e+202)))
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 3.3e+148], And[N[Not[LessEqual[y, 2.6e+163]], $MachinePrecision], LessEqual[y, 1.3e+202]]], 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 < 3.3000000000000001e148 or 2.6000000000000002e163 < y < 1.3000000000000001e202

   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 89.5%

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

   3. Taylor expanded in x around inf 74.2%

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

   if 3.3000000000000001e148 < y < 2.6000000000000002e163 or 1.3000000000000001e202 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-+l+ 99.4%

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

      4. *-commutative 99.4%

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

      5. distribute-rgt-neg-in 99.4%

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

      6. fma-def 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-neg-in 99.5%

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

      9. unsub-neg 99.5%

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

      10. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in z around 0 96.1%

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

      1. associate-+r+ 96.2%

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

      2. +-commutative 96.2%

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

      3. associate-*r* 96.2%

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

      4. neg-mul-1 96.2%

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

      5. +-commutative 96.2%

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

      6. cancel-sign-sub-inv 96.2%

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

      7. associate--l+ 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in y around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. log-rec 82.0%

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

      3. remove-double-neg 82.0%

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

   9. Simplified 82.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+148} \lor \neg \left(y \leq 2.6 \cdot 10^{+163}\right) \land y \leq 1.3 \cdot 10^{+202}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-247}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.9e+73)
   (- x z)
   (if (<= z 2.5e-247)
     (+ x (* y (- 1.0 (log y))))
     (if (<= z 5.5e-10) (- x (* (log y) 0.5)) (- x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.9e+73) {
		tmp = x - z;
	} else if (z <= 2.5e-247) {
		tmp = x + (y * (1.0 - log(y)));
	} else if (z <= 5.5e-10) {
		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 <= (-4.9d+73)) then
        tmp = x - z
    else if (z <= 2.5d-247) then
        tmp = x + (y * (1.0d0 - log(y)))
    else if (z <= 5.5d-10) 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 <= -4.9e+73) {
		tmp = x - z;
	} else if (z <= 2.5e-247) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else if (z <= 5.5e-10) {
		tmp = x - (Math.log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.9e+73:
		tmp = x - z
	elif z <= 2.5e-247:
		tmp = x + (y * (1.0 - math.log(y)))
	elif z <= 5.5e-10:
		tmp = x - (math.log(y) * 0.5)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.9e+73)
		tmp = Float64(x - z);
	elseif (z <= 2.5e-247)
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	elseif (z <= 5.5e-10)
		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 <= -4.9e+73)
		tmp = x - z;
	elseif (z <= 2.5e-247)
		tmp = x + (y * (1.0 - log(y)));
	elseif (z <= 5.5e-10)
		tmp = x - (log(y) * 0.5);
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.9e+73], N[(x - z), $MachinePrecision], If[LessEqual[z, 2.5e-247], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.5e-10], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8999999999999999e73 or 5.4999999999999996e-10 < z

   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 82.5%

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

   3. Taylor expanded in x around inf 82.5%

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

   if -4.8999999999999999e73 < z < 2.49999999999999989e-247

   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. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. unsub-neg 99.8%

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

      10. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 79.9%

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

      1. log-rec 79.9%

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

      2. sub-neg 79.9%

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

   6. Simplified 79.9%

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

   if 2.49999999999999989e-247 < z < 5.4999999999999996e-10

   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+ 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. unsub-neg 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. neg-mul-1 99.6%

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

      5. +-commutative 99.6%

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

      6. cancel-sign-sub-inv 99.6%

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

      7. associate--l+ 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around 0 86.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-247}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;t_0 - z\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-247}:\\ \;\;\;\;x + t_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= z -2.4e+22)
     (- t_0 z)
     (if (<= z 2.6e-247)
       (+ x t_0)
       (if (<= z 5.5e-10) (- x (* (log y) 0.5)) (- x z))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (z <= -2.4e+22) {
		tmp = t_0 - z;
	} else if (z <= 2.6e-247) {
		tmp = x + t_0;
	} else if (z <= 5.5e-10) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (z <= (-2.4d+22)) then
        tmp = t_0 - z
    else if (z <= 2.6d-247) then
        tmp = x + t_0
    else if (z <= 5.5d-10) 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 t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (z <= -2.4e+22) {
		tmp = t_0 - z;
	} else if (z <= 2.6e-247) {
		tmp = x + t_0;
	} else if (z <= 5.5e-10) {
		tmp = x - (Math.log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if z <= -2.4e+22:
		tmp = t_0 - z
	elif z <= 2.6e-247:
		tmp = x + t_0
	elif z <= 5.5e-10:
		tmp = x - (math.log(y) * 0.5)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (z <= -2.4e+22)
		tmp = Float64(t_0 - z);
	elseif (z <= 2.6e-247)
		tmp = Float64(x + t_0);
	elseif (z <= 5.5e-10)
		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)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (z <= -2.4e+22)
		tmp = t_0 - z;
	elseif (z <= 2.6e-247)
		tmp = x + t_0;
	elseif (z <= 5.5e-10)
		tmp = x - (log(y) * 0.5);
	else
		tmp = x - 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[z, -2.4e+22], N[(t$95$0 - z), $MachinePrecision], If[LessEqual[z, 2.6e-247], N[(x + t$95$0), $MachinePrecision], If[LessEqual[z, 5.5e-10], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.4e22

   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. add-sqr-sqrt 43.5%

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

      2. pow2 43.5%

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

      3. *-commutative 43.5%

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

   3. Applied egg-rr 43.5%

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

      1. unpow2 43.5%

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

      2. add-sqr-sqrt 99.9%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate-*r* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 91.8%

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

      1. log-rec 91.8%

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

      2. mul-1-neg 91.8%

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

      3. remove-double-neg 91.8%

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

   8. Simplified 91.8%

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

   if -2.4e22 < z < 2.6e-247

   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. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. unsub-neg 99.8%

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

      10. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 81.6%

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

      1. log-rec 81.6%

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

      2. sub-neg 81.6%

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

   6. Simplified 81.6%

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

   if 2.6e-247 < z < 5.4999999999999996e-10

   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+ 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. unsub-neg 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. neg-mul-1 99.6%

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

      5. +-commutative 99.6%

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

      6. cancel-sign-sub-inv 99.6%

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

      7. associate--l+ 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around 0 86.7%

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

   if 5.4999999999999996e-10 < z

   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 80.2%

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

   3. Taylor expanded in x around inf 80.2%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-247}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 5: 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 6: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+20} \lor \neg \left(z \leq 5.5 \cdot 10^{-10}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.1e+20) (not (<= z 5.5e-10))) (- x z) (- x (* (log y) 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.1e+20) || !(z <= 5.5e-10)) {
		tmp = x - z;
	} else {
		tmp = x - (log(y) * 0.5);
	}
	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 <= (-5.1d+20)) .or. (.not. (z <= 5.5d-10))) then
        tmp = x - z
    else
        tmp = x - (log(y) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.1e+20) || !(z <= 5.5e-10)) {
		tmp = x - z;
	} else {
		tmp = x - (Math.log(y) * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.1e+20) or not (z <= 5.5e-10):
		tmp = x - z
	else:
		tmp = x - (math.log(y) * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.1e+20) || !(z <= 5.5e-10))
		tmp = Float64(x - z);
	else
		tmp = Float64(x - Float64(log(y) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.1e+20) || ~((z <= 5.5e-10)))
		tmp = x - z;
	else
		tmp = x - (log(y) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.1e+20], N[Not[LessEqual[z, 5.5e-10]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1e20 or 5.4999999999999996e-10 < z

   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 81.0%

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

   3. Taylor expanded in x around inf 81.0%

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

   if -5.1e20 < z < 5.4999999999999996e-10

   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. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. unsub-neg 99.8%

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

      10. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. neg-mul-1 99.7%

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

      5. +-commutative 99.7%

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

      6. cancel-sign-sub-inv 99.7%

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

      7. associate--l+ 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around 0 69.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+20} \lor \neg \left(z \leq 5.5 \cdot 10^{-10}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \end{array} \]

Alternative 7: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{+121}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.3e+121) (- (- x (* (log y) 0.5)) z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.3e+121) {
		tmp = (x - (log(y) * 0.5)) - 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 <= 7.3d+121) then
        tmp = (x - (log(y) * 0.5d0)) - 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 <= 7.3e+121) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.3e+121:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.3e+121)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - 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 <= 7.3e+121)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.3e+121], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $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 < 7.3e121

   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 93.2%

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

   if 7.3e121 < 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+ 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

      6. fma-def 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-neg-in 99.6%

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

      9. unsub-neg 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 90.2%

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

      1. log-rec 90.2%

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

      2. sub-neg 90.2%

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

   6. Simplified 90.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.3 \cdot 10^{+121}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 8: 48.3% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9600000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+24}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9600000000.0) x (if (<= x 6.2e+24) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9600000000.0) {
		tmp = x;
	} else if (x <= 6.2e+24) {
		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 <= (-9600000000.0d0)) then
        tmp = x
    else if (x <= 6.2d+24) 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 <= -9600000000.0) {
		tmp = x;
	} else if (x <= 6.2e+24) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9600000000.0:
		tmp = x
	elif x <= 6.2e+24:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9600000000.0)
		tmp = x;
	elseif (x <= 6.2e+24)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9600000000.0)
		tmp = x;
	elseif (x <= 6.2e+24)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9600000000.0], x, If[LessEqual[x, 6.2e+24], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.6e9 or 6.20000000000000022e24 < 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. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. unsub-neg 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 71.0%

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

   if -9.6e9 < x < 6.20000000000000022e24

   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. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. unsub-neg 99.8%

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

      10. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-+r+ 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. *-commutative 99.8%

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

      9. sub-neg 99.8%

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

      10. add-cube-cbrt 98.7%

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

      11. fma-def 98.7%

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

   5. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2}, \sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}, y - z\right)} \]

   6. Taylor expanded in z around inf 44.9%

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

      1. neg-mul-1 44.9%

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

   8. Simplified 44.9%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9600000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+24}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 57.8% 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. Taylor expanded in y around 0 75.2%

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

3. Taylor expanded in x around inf 62.8%

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

4. Final simplification 62.8%

   \[\leadsto x - z \]

Alternative 10: 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. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 35.5%

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

5. Final simplification 35.5%

   \[\leadsto x \]

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 2023311 
(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))