Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 14.4s

Alternatives: 16

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   3. associate--l+ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. unsub-neg N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-+.f64 N/A

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

   18. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.43:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, x + \left(y - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.43)
   (fma (- -0.5 y) (log y) (- x z))
   (fma (- y) (log y) (+ x (- y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.43) {
		tmp = fma((-0.5 - y), log(y), (x - z));
	} else {
		tmp = fma(-y, log(y), (x + (y - z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.43)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x - z));
	else
		tmp = fma(Float64(-y), log(y), Float64(x + Float64(y - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.43], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[((-y) * N[Log[y], $MachinePrecision] + N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.429999999999999993

   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

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 0.429999999999999993 < 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. 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

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.43:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.43)
   (fma (- -0.5 y) (log y) (- x z))
   (- (+ y (- x (* y (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.43) {
		tmp = fma((-0.5 - y), log(y), (x - z));
	} else {
		tmp = (y + (x - (y * log(y)))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.43)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(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, 0.43], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $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 < 0.429999999999999993

   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

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 98.7

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

   7. Applied rewrites 98.7%

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

   if 0.429999999999999993 < 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

      \[\leadsto \left(\left(x - \color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}\right) + y\right) - z \]
   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. Applied rewrites 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 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.43:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+53}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.8e+53) (- z) (if (<= z 1.1e+59) (fma -0.5 (log y) x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e+53) {
		tmp = -z;
	} else if (z <= 1.1e+59) {
		tmp = fma(-0.5, log(y), x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.8e+53)
		tmp = Float64(-z);
	elseif (z <= 1.1e+59)
		tmp = fma(-0.5, log(y), x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.8e+53], (-z), If[LessEqual[z, 1.1e+59], N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.79999999999999997e53 or 1.1e59 < z

   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 59.3

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

   5. Applied rewrites 59.3%

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

   if -3.79999999999999997e53 < z < 1.1e59

   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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 64.5%

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

Alternative 5: 90.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+119)
   (fma (- -0.5 y) (log y) (- x z))
   (fma (- -0.5 y) (log y) (- y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+119) {
		tmp = fma((-0.5 - y), log(y), (x - z));
	} else {
		tmp = fma((-0.5 - y), log(y), (y - z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+119)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x - z));
	else
		tmp = fma(Float64(-0.5 - y), log(y), Float64(y - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+119], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e119

   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

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 94.3

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

   7. Applied rewrites 94.3%

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

   if 4.9999999999999999e119 < y

   1. Initial program 99.5%

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

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 91.5

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

   7. Applied rewrites 91.5%

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

Alternative 6: 90.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+119)
   (fma (- -0.5 y) (log y) (- x z))
   (- (fma (log y) (- y) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+119) {
		tmp = fma((-0.5 - y), log(y), (x - z));
	} else {
		tmp = fma(log(y), -y, y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+119)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x - z));
	else
		tmp = Float64(fma(log(y), Float64(-y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+119], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e119

   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

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 94.3

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

   7. Applied rewrites 94.3%

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

   if 4.9999999999999999e119 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

Alternative 7: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.95e+71)
   (- (fma (log y) -0.5 x) z)
   (- (fma (log y) (- -0.5 y) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e+71) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = fma(log(y), (-0.5 - y), y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.95e+71)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(fma(log(y), Float64(-0.5 - y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.95e+71], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9500000000000001e71

   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 96.5

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

   5. Applied rewrites 96.5%

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

   if 1.9500000000000001e71 < 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

      \[\leadsto \color{blue}{\left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} - z \]
   4. 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 86.5

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

   5. Applied rewrites 86.5%

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

Alternative 8: 42.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1900:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 260:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1900.0) (- z) (if (<= z 260.0) (* -0.5 (log y)) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1900.0) {
		tmp = -z;
	} else if (z <= 260.0) {
		tmp = -0.5 * log(y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1900.0d0)) then
        tmp = -z
    else if (z <= 260.0d0) then
        tmp = (-0.5d0) * log(y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1900.0) {
		tmp = -z;
	} else if (z <= 260.0) {
		tmp = -0.5 * Math.log(y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1900.0:
		tmp = -z
	elif z <= 260.0:
		tmp = -0.5 * math.log(y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1900.0)
		tmp = Float64(-z);
	elseif (z <= 260.0)
		tmp = Float64(-0.5 * log(y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1900.0)
		tmp = -z;
	elseif (z <= 260.0)
		tmp = -0.5 * log(y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1900.0], (-z), If[LessEqual[z, 260.0], N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1900 or 260 < z

   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 52.9

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

   5. Applied rewrites 52.9%

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

   if -1900 < z < 260

   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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{-1}{2} \cdot \log y \]
   10. Step-by-step derivation

   11. Applied rewrites 26.4%

       \[\leadsto -0.5 \cdot \log y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.95e+71) (- (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 <= 1.95e+71) {
		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 <= 1.95e+71)
		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, 1.95e+71], 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 < 1.9500000000000001e71

   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 96.5

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

   5. Applied rewrites 96.5%

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

   if 1.9500000000000001e71 < 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

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} - z \]
   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 86.5

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

   5. Applied rewrites 86.5%

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

Alternative 10: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.2e+131) (- (fma (log y) -0.5 x) z) (+ y (fma (log y) (- y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.2e+131) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y + fma(log(y), -y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.2e+131)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y + fma(log(y), Float64(-y), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.2e+131], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y + N[(N[Log[y], $MachinePrecision] * (-y) + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e131

   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

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

   if 1.2e131 < 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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 86.7%

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

Alternative 11: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e+131) (- (fma (log y) -0.5 x) z) (fma (log y) (- y) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e+131) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = fma(log(y), -y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7e+131)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = fma(log(y), Float64(-y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7e+131], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.9999999999999998e131

   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

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

   if 6.9999999999999998e131 < 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 78.3

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

   5. Applied rewrites 78.3%

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

Alternative 12: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. +-commutative N/A

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

   4. lift--.f64 N/A

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

   5. associate-+r- N/A

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

   6. associate--l- N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 13: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. associate--l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. lower-fma.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. distribute-neg-in N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 14: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.95e+71) (fma -0.5 (log y) x) (fma (log y) (- y) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e+71) {
		tmp = fma(-0.5, log(y), x);
	} else {
		tmp = fma(log(y), -y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.95e+71)
		tmp = fma(-0.5, log(y), x);
	else
		tmp = fma(log(y), Float64(-y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.95e+71], N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9500000000000001e71

   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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.4%

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

   if 1.9500000000000001e71 < 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 66.3

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

   5. Applied rewrites 66.3%

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

Alternative 15: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.95e+71) (fma -0.5 (log y) x) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e+71) {
		tmp = fma(-0.5, log(y), x);
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.95e+71)
		tmp = fma(-0.5, log(y), x);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.95e+71], N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9500000000000001e71

   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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.4%

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

   if 1.9500000000000001e71 < 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 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 21.5

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

   5. Applied rewrites 21.5%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. log-rec N/A

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

      3. cancel-sign-sub N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

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

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 66.3

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

   8. Applied rewrites 66.3%

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

Alternative 16: 29.7% 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 27.0

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

5. Applied rewrites 27.0%

   \[\leadsto \color{blue}{-z} \]
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 2024220 
(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))