Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 7.7s

Alternatives: 15

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 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 73.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+166}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq -6 \cdot 10^{+44}:\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;y - \log y \cdot y\\ \mathbf{elif}\;t\_0 \leq 341.25:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -2e+166)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 -6e+44)
       (- (+ (pow (pow x -1.0) -1.0) y) z)
       (if (<= t_0 -5e+16)
         (- y (* (log y) y))
         (if (<= t_0 341.25) (- (* -0.5 (log y)) z) (fma (/ (- z) x) x x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double tmp;
	if (t_0 <= -2e+166) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= -6e+44) {
		tmp = (pow(pow(x, -1.0), -1.0) + y) - z;
	} else if (t_0 <= -5e+16) {
		tmp = y - (log(y) * y);
	} else if (t_0 <= 341.25) {
		tmp = (-0.5 * log(y)) - z;
	} else {
		tmp = fma((-z / x), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	tmp = 0.0
	if (t_0 <= -2e+166)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= -6e+44)
		tmp = Float64(Float64(((x ^ -1.0) ^ -1.0) + y) - z);
	elseif (t_0 <= -5e+16)
		tmp = Float64(y - Float64(log(y) * y));
	elseif (t_0 <= 341.25)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	else
		tmp = fma(Float64(Float64(-z) / x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+166], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, -6e+44], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], -1.0], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, -5e+16], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 341.25], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.99999999999999988e166

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-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 76.4

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

   5. Applied rewrites 76.4%

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

   if -1.99999999999999988e166 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -5.99999999999999974e44

   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

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if -5.99999999999999974e44 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -5e16

   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 84.4

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

   5. Applied rewrites 84.4%

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

   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 64.9%

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

   if -5e16 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 341.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 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 98.2

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

   5. Applied rewrites 98.2%

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

   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 96.3%

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

   if 341.25 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

      \[\leadsto \mathsf{fma}\left(\frac{-z}{x}, x, x\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -2 \cdot 10^{+166}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -6 \cdot 10^{+44}:\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -5 \cdot 10^{+16}:\\ \;\;\;\;y - \log y \cdot y\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq 341.25:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ t_1 := \left(1 - \log y\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -6 \cdot 10^{+44}:\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 341.25:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)) (t_1 (* (- 1.0 (log y)) y)))
   (if (<= t_0 -2e+166)
     t_1
     (if (<= t_0 -6e+44)
       (- (+ (pow (pow x -1.0) -1.0) y) z)
       (if (<= t_0 -5e+16)
         t_1
         (if (<= t_0 341.25) (- (* -0.5 (log y)) z) (fma (/ (- z) x) x x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double t_1 = (1.0 - log(y)) * y;
	double tmp;
	if (t_0 <= -2e+166) {
		tmp = t_1;
	} else if (t_0 <= -6e+44) {
		tmp = (pow(pow(x, -1.0), -1.0) + y) - z;
	} else if (t_0 <= -5e+16) {
		tmp = t_1;
	} else if (t_0 <= 341.25) {
		tmp = (-0.5 * log(y)) - z;
	} else {
		tmp = fma((-z / x), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	t_1 = Float64(Float64(1.0 - log(y)) * y)
	tmp = 0.0
	if (t_0 <= -2e+166)
		tmp = t_1;
	elseif (t_0 <= -6e+44)
		tmp = Float64(Float64(((x ^ -1.0) ^ -1.0) + y) - z);
	elseif (t_0 <= -5e+16)
		tmp = t_1;
	elseif (t_0 <= 341.25)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	else
		tmp = fma(Float64(Float64(-z) / x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+166], t$95$1, If[LessEqual[t$95$0, -6e+44], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], -1.0], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, -5e+16], t$95$1, If[LessEqual[t$95$0, 341.25], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.99999999999999988e166 or -5.99999999999999974e44 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -5e16

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-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 74.6

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

   5. Applied rewrites 74.6%

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

   if -1.99999999999999988e166 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -5.99999999999999974e44

   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

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 58.7

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

   7. Applied rewrites 58.7%

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

   if -5e16 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 341.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 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 98.2

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

   5. Applied rewrites 98.2%

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

   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 96.3%

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

   if 341.25 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

      \[\leadsto \mathsf{fma}\left(\frac{-z}{x}, x, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -2 \cdot 10^{+166}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -6 \cdot 10^{+44}:\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq 341.25:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return (math.pow(math.pow(x, -1.0), -1.0) + y) - z

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], -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

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 99.7

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 51.8

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

7. Applied rewrites 51.8%

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

8. Final simplification 51.8%

   \[\leadsto \left({\left({x}^{-1}\right)}^{-1} + y\right) - z \]
9. Add Preprocessing

Alternative 5: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -66000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right) - z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -66000000.0)
   (- (fma (- -0.5 y) (log y) y) z)
   (if (<= z 2.7e+74)
     (fma (- -0.5 y) (log y) (+ y x))
     (- (fma (- y) (log y) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -66000000.0) {
		tmp = fma((-0.5 - y), log(y), y) - z;
	} else if (z <= 2.7e+74) {
		tmp = fma((-0.5 - y), log(y), (y + x));
	} else {
		tmp = fma(-y, log(y), y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -66000000.0)
		tmp = Float64(fma(Float64(-0.5 - y), log(y), y) - z);
	elseif (z <= 2.7e+74)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(y + x));
	else
		tmp = Float64(fma(Float64(-y), log(y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -66000000.0], N[(N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 2.7e+74], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[((-y) * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.6e7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 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. *-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. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if -6.6e7 < z < 2.6999999999999998e74

   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 5.9

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if 2.6999999999999998e74 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.9

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      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. associate-*l* N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-log.f64 91.0

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

   7. Applied rewrites 91.0%

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

   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 91.0%

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

Alternative 6: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -66000000:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -66000000.0)
   (- y (fma (+ 0.5 y) (log y) z))
   (if (<= z 2.7e+74)
     (fma (- -0.5 y) (log y) (+ y x))
     (- (fma (- y) (log y) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -66000000.0) {
		tmp = y - fma((0.5 + y), log(y), z);
	} else if (z <= 2.7e+74) {
		tmp = fma((-0.5 - y), log(y), (y + x));
	} else {
		tmp = fma(-y, log(y), y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -66000000.0)
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	elseif (z <= 2.7e+74)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(y + x));
	else
		tmp = Float64(fma(Float64(-y), log(y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -66000000.0], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e+74], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[((-y) * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.6e7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 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 92.0

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

   5. Applied rewrites 92.0%

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

   if -6.6e7 < z < 2.6999999999999998e74

   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 5.9

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if 2.6999999999999998e74 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.9

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      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. associate-*l* N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-log.f64 91.0

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

   7. Applied rewrites 91.0%

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

   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 91.0%

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

Alternative 7: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -48 \lor \neg \left(x \leq 4800000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -48.0) (not (<= x 4800000000.0)))
   (fma (/ (- z) x) x x)
   (- (* -0.5 (log y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -48.0) || !(x <= 4800000000.0)) {
		tmp = fma((-z / x), x, x);
	} else {
		tmp = (-0.5 * log(y)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -48.0) || !(x <= 4800000000.0))
		tmp = fma(Float64(Float64(-z) / x), x, x);
	else
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -48.0], N[Not[LessEqual[x, 4800000000.0]], $MachinePrecision]], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -48 or 4.8e9 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 72.1%

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

   if -48 < x < 4.8e9

   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 99.2

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

   5. Applied rewrites 99.2%

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

   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 62.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48 \lor \neg \left(x \leq 4800000000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(-y, \log y, y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e-15)
   (- (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 <= 6.5e-15) {
		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 <= 6.5e-15)
		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, 6.5e-15], 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 < 6.49999999999999991e-15

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 6.49999999999999991e-15 < 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

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

      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. 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 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 9: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 135:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 135.0) (- (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 <= 135.0) {
		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 <= 135.0)
		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, 135.0], 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 < 135

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 135 < 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 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 83.5

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

   5. Applied rewrites 83.5%

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

Alternative 10: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.95e+15) (- (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 <= 1.95e+15) {
		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 <= 1.95e+15)
		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, 1.95e+15], 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 < 1.95e15

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 1.95e15 < 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

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      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. associate-*l* N/A

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

      13. lft-mult-inverse N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-log.f64 83.8

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

   7. Applied rewrites 83.8%

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

   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 83.8%

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

Alternative 11: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2100000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2100000000000.0)
   (- (fma -0.5 (log y) x) z)
   (fma (- y) (log y) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2100000000000.0) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(-y, log(y), (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2100000000000.0)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = fma(Float64(-y), log(y), Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2100000000000.0], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[((-y) * N[Log[y], $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e12

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 2.1e12 < 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 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 17.2

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

   5. Applied rewrites 17.2%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 83.8

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 83.8%

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

Alternative 12: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e+137) (- (fma -0.5 (log y) x) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+137) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e+137)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.3e+137], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e137

   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 87.0

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

   5. Applied rewrites 87.0%

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

   if 1.3e137 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

Alternative 13: 57.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-95} \lor \neg \left(x \leq 4.5 \cdot 10^{-47}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.8e-95) (not (<= x 4.5e-47))) (fma (/ (- z) x) x x) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.8e-95) || !(x <= 4.5e-47)) {
		tmp = fma((-z / x), x, x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.8e-95) || !(x <= 4.5e-47))
		tmp = fma(Float64(Float64(-z) / x), x, x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.8e-95], N[Not[LessEqual[x, 4.5e-47]], $MachinePrecision]], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999997e-95 or 4.5e-47 < x

   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 inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 63.3%

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

   if -3.7999999999999997e-95 < x < 4.5e-47

   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 38.6

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

   5. Applied rewrites 38.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-95} \lor \neg \left(x \leq 4.5 \cdot 10^{-47}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -66000000 \lor \neg \left(z \leq 2.6 \cdot 10^{+74}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -66000000.0) (not (<= z 2.6e+74))) (- z) (* (/ x z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -66000000.0) || !(z <= 2.6e+74)) {
		tmp = -z;
	} else {
		tmp = (x / z) * 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 <= (-66000000.0d0)) .or. (.not. (z <= 2.6d+74))) then
        tmp = -z
    else
        tmp = (x / z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -66000000.0) || !(z <= 2.6e+74)) {
		tmp = -z;
	} else {
		tmp = (x / z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -66000000.0) or not (z <= 2.6e+74):
		tmp = -z
	else:
		tmp = (x / z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -66000000.0) || !(z <= 2.6e+74))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x / z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -66000000.0) || ~((z <= 2.6e+74)))
		tmp = -z;
	else
		tmp = (x / z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -66000000.0], N[Not[LessEqual[z, 2.6e+74]], $MachinePrecision]], (-z), N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6e7 or 2.6000000000000001e74 < 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 66.1

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

   5. Applied rewrites 66.1%

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

   if -6.6e7 < z < 2.6000000000000001e74

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 24.4%

       \[\leadsto \frac{x}{z} \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -66000000 \lor \neg \left(z \leq 2.6 \cdot 10^{+74}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.1% 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 28.9

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

5. Applied rewrites 28.9%

   \[\leadsto \color{blue}{-z} \]
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 2024318 
(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))