Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.8s

Alternatives: 11

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, 1.0× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift--.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{elif}\;t\_0 \leq 354:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -1000.0)
     (- y (fma y (log y) z))
     (if (<= t_0 354.0) (- (* (log y) -0.5) z) (fma x (/ z (- x)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = y - fma(y, log(y), z);
	} else if (t_0 <= 354.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = fma(x, (z / -x), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = Float64(y - fma(y, log(y), z));
	elseif (t_0 <= 354.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = fma(x, Float64(z / Float64(-x)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 354.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(z / (-x)), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1e3

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 72.6

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

   5. Simplified 72.6%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. log-rec N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-out-- N/A

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

      7. *-rgt-identity N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-out N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 72.5

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

   8. Simplified 72.5%

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

   if -1e3 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 354

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      6. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      9. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      10. flip3-+ N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} - z \]

      5. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) - z \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) - z \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) - z \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) - z \]

      9. lower--.f64 98.4

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

   7. Simplified 98.4%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y} - z \]

      3. lower-log.f64 98.4

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

   10. Simplified 98.4%

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

   if 354 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. div-sub N/A

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

      7. associate--r+ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{neg}\left(z\right)}}{x}, x\right) \]

      2. lower-neg.f64 98.3

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]

   8. Simplified 98.3%

      \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -1000:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 354:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;t\_0 \leq 354:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -2e+119)
     (fma (log y) (- y) y)
     (if (<= t_0 354.0) (- (* (log y) -0.5) z) (fma x (/ z (- x)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -2e+119) {
		tmp = fma(log(y), -y, y);
	} else if (t_0 <= 354.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = fma(x, (z / -x), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -2e+119)
		tmp = fma(log(y), Float64(-y), y);
	elseif (t_0 <= 354.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = fma(x, Float64(z / Float64(-x)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+119], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t$95$0, 354.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(z / (-x)), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.99999999999999989e119

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\mathsf{neg}\left(y\right)}, y\right) \]

      14. lower-neg.f64 57.4

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

   5. Simplified 57.4%

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

   if -1.99999999999999989e119 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 354

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      6. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      9. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      10. flip3-+ N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} - z \]

      5. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) - z \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) - z \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) - z \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) - z \]

      9. lower--.f64 89.1

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

   7. Simplified 89.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y} - z \]

      3. lower-log.f64 77.4

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

   10. Simplified 77.4%

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

   if 354 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. div-sub N/A

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

      7. associate--r+ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{neg}\left(z\right)}}{x}, x\right) \]

      2. lower-neg.f64 98.3

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]

   8. Simplified 98.3%

      \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 354:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+119}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;t\_0 \leq 354:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -2e+119)
     (- y (* y (log y)))
     (if (<= t_0 354.0) (- (* (log y) -0.5) z) (fma x (/ z (- x)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -2e+119) {
		tmp = y - (y * log(y));
	} else if (t_0 <= 354.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = fma(x, (z / -x), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -2e+119)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (t_0 <= 354.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = fma(x, Float64(z / Float64(-x)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+119], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 354.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(z / (-x)), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.99999999999999989e119

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\mathsf{neg}\left(y\right)}, y\right) \]

      14. lower-neg.f64 57.4

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

   5. Simplified 57.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -y, y\right)} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 57.3

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

   7. Applied egg-rr 57.3%

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

   if -1.99999999999999989e119 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 354

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      6. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      9. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      10. flip3-+ N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} - z \]

      5. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) - z \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) - z \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) - z \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) - z \]

      9. lower--.f64 89.1

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

   7. Simplified 89.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y} - z \]

      3. lower-log.f64 77.4

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

   10. Simplified 77.4%

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

   if 354 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. div-sub N/A

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

      7. associate--r+ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{neg}\left(z\right)}}{x}, x\right) \]

      2. lower-neg.f64 98.3

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]

   8. Simplified 98.3%

      \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -2 \cdot 10^{+119}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 354:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.4e-8) (- (fma (log y) -0.5 x) z) (- (+ y (- x (* y (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.4e-8) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = (y + (x - (y * log(y)))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.4e-8)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(Float64(y + Float64(x - Float64(y * log(y)))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.4e-8], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. lower-log.f64 100.0

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

   5. Simplified 100.0%

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

   if 3.4e-8 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 99.2

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

   5. Simplified 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.8e+87) (- (fma (log y) -0.5 x) z) (- (fma (log y) (- y) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.8e+87) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = fma(log(y), -y, y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.8e+87)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(fma(log(y), Float64(-y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.8e+87], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999997e87

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. lower-log.f64 94.5

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

   5. Simplified 94.5%

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

   if 1.79999999999999997e87 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 87.7

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

   5. Simplified 87.7%

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

Alternative 7: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.8e+87) (- (fma (log y) -0.5 x) z) (- y (fma y (log y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.8e+87) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y - fma(y, log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.8e+87)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y - fma(y, log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.8e+87], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999997e87

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. lower-log.f64 94.5

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

   5. Simplified 94.5%

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

   if 1.79999999999999997e87 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 87.7

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

   5. Simplified 87.7%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. log-rec N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-out-- N/A

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

      7. *-rgt-identity N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-out N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 87.5

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

   8. Simplified 87.5%

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

Alternative 8: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+151) (fma x (/ z (- x)) x) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+151) {
		tmp = fma(x, (z / -x), x);
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+151)
		tmp = fma(x, Float64(z / Float64(-x)), x);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+151], N[(x * N[(z / (-x)), $MachinePrecision] + x), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000003e151

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. div-sub N/A

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

      7. associate--r+ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Simplified 88.5%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{neg}\left(z\right)}}{x}, x\right) \]

      2. lower-neg.f64 64.3

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]

   8. Simplified 64.3%

      \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]

   if 2.00000000000000003e151 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\mathsf{neg}\left(y\right)}, y\right) \]

      14. lower-neg.f64 74.9

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

   5. Simplified 74.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -y, y\right)} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.8

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

   7. Applied egg-rr 74.8%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \mathbf{if}\;x \leq -44:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x (/ z (- x)) x)))
   (if (<= x -44.0) t_0 (if (<= x 1.2e-61) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, (z / -x), x);
	double tmp;
	if (x <= -44.0) {
		tmp = t_0;
	} else if (x <= 1.2e-61) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, Float64(z / Float64(-x)), x)
	tmp = 0.0
	if (x <= -44.0)
		tmp = t_0;
	elseif (x <= 1.2e-61)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(z / (-x)), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -44.0], t$95$0, If[LessEqual[x, 1.2e-61], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -44 or 1.2e-61 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. div-sub N/A

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

      7. associate--r+ N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{neg}\left(z\right)}}{x}, x\right) \]

      2. lower-neg.f64 75.9

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]

   8. Simplified 75.9%

      \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{-z}}{x}, x\right) \]

   if -44 < x < 1.2e-61

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 42.6

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

   5. Simplified 42.6%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -44:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{-x}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+80}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+78}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e+80) x (if (<= x 1.15e+78) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+80) {
		tmp = x;
	} else if (x <= 1.15e+78) {
		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 <= (-1.3d+80)) then
        tmp = x
    else if (x <= 1.15d+78) 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 <= -1.3e+80) {
		tmp = x;
	} else if (x <= 1.15e+78) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e+80:
		tmp = x
	elif x <= 1.15e+78:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e+80)
		tmp = x;
	elseif (x <= 1.15e+78)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.3e+80)
		tmp = x;
	elseif (x <= 1.15e+78)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e+80], x, If[LessEqual[x, 1.15e+78], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.29999999999999991e80 or 1.1500000000000001e78 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. associate--r+ N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\frac{\frac{1}{2} + y}{z}\right), \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right)} \]

      8. lower-log.f64 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\frac{\frac{1}{2} + y}{z}\right), \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z \cdot \mathsf{fma}\left(\log y, \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)}{z}}, \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto z \cdot \mathsf{fma}\left(\log y, \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)}{z}}, \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

      11. distribute-neg-in N/A

          \[\leadsto z \cdot \mathsf{fma}\left(\log y, \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}}{z}, \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

      12. metadata-eval N/A

          \[\leadsto z \cdot \mathsf{fma}\left(\log y, \frac{\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right)}{z}, \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. sub-neg N/A

          \[\leadsto z \cdot \mathsf{fma}\left(\log y, \frac{\frac{-1}{2} - y}{z}, \color{blue}{\left(\frac{x}{z} + \frac{y}{z}\right) + \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   5. Simplified 76.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 52.8

         \[\leadsto z \cdot \color{blue}{\frac{x}{z}} \]

   8. Simplified 52.8%

      \[\leadsto z \cdot \color{blue}{\frac{x}{z}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{x}{z}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot z} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot z \]

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

         \[\leadsto x \cdot {z}^{\color{blue}{0}} \]

      9. metadata-eval N/A

         \[\leadsto x \cdot \color{blue}{1} \]

      10. lower-*.f64 72.3

          \[\leadsto \color{blue}{x \cdot 1} \]

   10. Applied egg-rr 72.3%

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

       1. *-rgt-identity 72.3

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

   12. Applied egg-rr 72.3%

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

   if -1.29999999999999991e80 < x < 1.1500000000000001e78

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 41.1

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

   5. Simplified 41.1%

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

Alternative 11: 29.9% accurate, 118.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. associate--r+ N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\frac{\frac{1}{2} + y}{z}\right), \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right)} \]

   8. lower-log.f64 N/A

      \[\leadsto z \cdot \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\frac{\frac{1}{2} + y}{z}\right), \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto z \cdot \mathsf{fma}\left(\log y, \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)}{z}}, \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

   10. lower-/.f64 N/A

       \[\leadsto z \cdot \mathsf{fma}\left(\log y, \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)}{z}}, \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

   11. distribute-neg-in N/A

       \[\leadsto z \cdot \mathsf{fma}\left(\log y, \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}}{z}, \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

   12. metadata-eval N/A

       \[\leadsto z \cdot \mathsf{fma}\left(\log y, \frac{\color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right)}{z}, \left(\frac{x}{z} + \frac{y}{z}\right) - 1\right) \]

   13. unsub-neg N/A

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

   14. lower--.f64 N/A

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

   15. sub-neg N/A

       \[\leadsto z \cdot \mathsf{fma}\left(\log y, \frac{\frac{-1}{2} - y}{z}, \color{blue}{\left(\frac{x}{z} + \frac{y}{z}\right) + \left(\mathsf{neg}\left(1\right)\right)}\right) \]

5. Simplified 84.3%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 25.7

      \[\leadsto z \cdot \color{blue}{\frac{x}{z}} \]

8. Simplified 25.7%

   \[\leadsto z \cdot \color{blue}{\frac{x}{z}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto z \cdot \color{blue}{\frac{x}{z}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x}{z} \cdot z} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot z \]

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. inv-pow N/A

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

   7. pow-plus N/A

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

   8. metadata-eval N/A

      \[\leadsto x \cdot {z}^{\color{blue}{0}} \]

   9. metadata-eval N/A

      \[\leadsto x \cdot \color{blue}{1} \]

   10. lower-*.f64 33.7

       \[\leadsto \color{blue}{x \cdot 1} \]

10. Applied egg-rr 33.7%

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

    1. *-rgt-identity 33.7

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

12. Applied egg-rr 33.7%

    \[\leadsto \color{blue}{x} \]
13. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- (+ y x) z) (* (+ y 1/2) (log y))))

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