Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.3s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. associate-+l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 88.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00045:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.00045)
   (- y (fma (+ 0.5 y) (log y) z))
   (if (<= z 2e+110)
     (fma (- -0.5 y) (log y) (+ x y))
     (- (fma -0.5 (log y) x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.00045) {
		tmp = y - fma((0.5 + y), log(y), z);
	} else if (z <= 2e+110) {
		tmp = fma((-0.5 - y), log(y), (x + y));
	} else {
		tmp = fma(-0.5, log(y), x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.00045)
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	elseif (z <= 2e+110)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x + y));
	else
		tmp = Float64(fma(-0.5, log(y), x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.00045], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+110], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.4999999999999999e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -4.4999999999999999e-4 < z < 2e110

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.5

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. lft-mult-inverse N/A

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

      10. associate-*l* N/A

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

      11. distribute-neg-in N/A

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

      12. mul-1-neg N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. lower--.f64 N/A

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

      21. lower-log.f64 N/A

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

      22. lower-+.f64 96.9

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

   7. Applied rewrites 96.9%

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

   if 2e110 < z

   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

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 95.8

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

   5. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \log y, x\right) - z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -57000000 \lor \neg \left(x \leq 21000\right):\\ \;\;\;\;\left(\left(-x\right) \cdot -1 + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -57000000.0) (not (<= x 21000.0)))
   (- (+ (* (- x) -1.0) y) z)
   (fma -0.5 (log y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -57000000.0) || !(x <= 21000.0)) {
		tmp = ((-x * -1.0) + y) - z;
	} else {
		tmp = fma(-0.5, log(y), -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -57000000.0) || !(x <= 21000.0))
		tmp = Float64(Float64(Float64(Float64(-x) * -1.0) + y) - z);
	else
		tmp = fma(-0.5, log(y), Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -57000000.0], N[Not[LessEqual[x, 21000.0]], $MachinePrecision]], N[(N[(N[((-x) * -1.0), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.7e7 or 21000 < 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. 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. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]
   9. Step-by-step derivation

   10. Applied rewrites 73.6%

       \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]

   if -5.7e7 < x < 21000

   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 x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -57000000 \lor \neg \left(x \leq 21000\right):\\ \;\;\;\;\left(\left(-x\right) \cdot -1 + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(-y, \log y, y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.85e-8)
   (- (fma -0.5 (log y) x) z)
   (- (+ x (fma (- y) (log y) y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.85e-8) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (x + fma(-y, log(y), y)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.85e-8)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(x + fma(Float64(-y), log(y), y)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.85e-8], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + N[((-y) * N[Log[y], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.85000000000000004e-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

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 2.85000000000000004e-8 < y

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.1

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

   7. Applied rewrites 99.1%

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

Alternative 5: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 10000000000.0)
   (- (fma -0.5 (log y) x) z)
   (- y (fma (+ 0.5 y) (log y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 10000000000.0) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - fma((0.5 + y), log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 10000000000.0)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 10000000000.0], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e10

   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

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1e10 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 88.7

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

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{y - \mathsf{fma}\left(0.5 + y, \log y, z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 74000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 74000000000.0)
   (- (fma -0.5 (log y) x) z)
   (- (fma (- y) (log y) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 74000000000.0) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(-y, log(y), y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 74000000000.0)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(fma(Float64(-y), log(y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 74000000000.0], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[((-y) * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.4e10

   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

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 7.4e10 < y

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

   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. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lft-mult-inverse N/A

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

      8. associate-*l* N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. mul-1-neg N/A

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

      20. unsub-neg N/A

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

      21. lower--.f64 N/A

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

      22. lower-log.f64 88.7

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

   7. Applied rewrites 88.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 88.6%

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

Alternative 7: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(0.5 + y\right) \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.6e+112) (- (fma -0.5 (log y) x) z) (- y (* (+ 0.5 y) (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e+112) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - ((0.5 + y) * log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.6e+112)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - Float64(Float64(0.5 + y) * log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.6e+112], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.59999999999999993e112

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if 1.59999999999999993e112 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 75.9%

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

Alternative 8: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.6e+112) (- (fma -0.5 (log y) x) z) (- y (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.6e+112) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - (log(y) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.6e+112)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - Float64(log(y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.6e+112], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.59999999999999993e112

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if 1.59999999999999993e112 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 75.9%

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

Alternative 9: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+77}:\\ \;\;\;\;\left(\left(-x\right) \cdot -1 + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+77) (- (+ (* (- x) -1.0) y) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+77) {
		tmp = ((-x * -1.0) + y) - z;
	} else {
		tmp = (1.0 - log(y)) * 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 <= 9d+77) then
        tmp = ((-x * (-1.0d0)) + y) - z
    else
        tmp = (1.0d0 - log(y)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+77) {
		tmp = ((-x * -1.0) + y) - z;
	} else {
		tmp = (1.0 - Math.log(y)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9e+77:
		tmp = ((-x * -1.0) + y) - z
	else:
		tmp = (1.0 - math.log(y)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e+77)
		tmp = Float64(Float64(Float64(Float64(-x) * -1.0) + y) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9e+77)
		tmp = ((-x * -1.0) + y) - z;
	else
		tmp = (1.0 - log(y)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e+77], N[(N[(N[((-x) * -1.0), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.00000000000000049e77

   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. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 100.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sub-neg N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]
   9. Step-by-step derivation

   10. Applied rewrites 77.2%

       \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]

   if 9.00000000000000049e77 < 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. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 71.7

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

   5. Applied rewrites 71.7%

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

Alternative 10: 57.7% accurate, 8.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((-x * -1.0) + y) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((-x * (-1.0d0)) + y) - z
end function

Java

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

Python

def code(x, y, z):
	return ((-x * -1.0) + y) - z

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing
3. 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. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-+ N/A

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

   10. lift-+.f64 N/A

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

   11. lower-/.f64 99.7

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. sub-neg N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-log.f64 86.1

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

7. Applied rewrites 86.1%

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

8. Taylor expanded in x around inf

   \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]
9. Step-by-step derivation

10. Applied rewrites 56.2%

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

Alternative 11: 30.0% accurate, 39.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 31.9

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

5. Applied rewrites 31.9%

   \[\leadsto \color{blue}{-z} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024324 
(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))