Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Percentage Accurate: 93.9% → 99.6%

Time: 15.0s

Alternatives: 21

Speedup: 0.9×

• 31.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, y + 0.0007936500793651, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - \left(x - \frac{0.083333333333333}{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4e-144)
   (fma
    (fma
     (fma z (+ y 0.0007936500793651) -0.0027777777777778)
     z
     0.083333333333333)
    (pow x -1.0)
    (fma (log x) (- x 0.5) (fma -1.0 x 0.91893853320467)))
   (fma
    (fma (/ z x) (+ y 0.0007936500793651) (/ -0.0027777777777778 x))
    z
    (fma
     (log x)
     (- x 0.5)
     (- 0.91893853320467 (- x (/ 0.083333333333333 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4e-144) {
		tmp = fma(fma(fma(z, (y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), pow(x, -1.0), fma(log(x), (x - 0.5), fma(-1.0, x, 0.91893853320467)));
	} else {
		tmp = fma(fma((z / x), (y + 0.0007936500793651), (-0.0027777777777778 / x)), z, fma(log(x), (x - 0.5), (0.91893853320467 - (x - (0.083333333333333 / x)))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4e-144)
		tmp = fma(fma(fma(z, Float64(y + 0.0007936500793651), -0.0027777777777778), z, 0.083333333333333), (x ^ -1.0), fma(log(x), Float64(x - 0.5), fma(-1.0, x, 0.91893853320467)));
	else
		tmp = fma(fma(Float64(z / x), Float64(y + 0.0007936500793651), Float64(-0.0027777777777778 / x)), z, fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - Float64(x - Float64(0.083333333333333 / x)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4e-144], N[(N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision] + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(-1.0 * x + 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - N[(x - N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999998e-144

   1. Initial program 99.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}\right) \cdot \frac{1}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}, \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z, y + \frac{7936500793651}{10000000000000000}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{y + \frac{7936500793651}{10000000000000000}}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} + y}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000} + y}, \mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \color{blue}{\frac{-13888888888889}{5000000000000000}}\right), z, \frac{83333333333333}{1000000000000000}\right), \frac{1}{x}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      18. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000} + y, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right), \color{blue}{{x}^{-1}}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      19. lower-pow.f64 99.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), \color{blue}{{x}^{-1}}, \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651 + y, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)} \]

   if 3.9999999999999998e-144 < x

   1. Initial program 91.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   5. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \left(\frac{\frac{83333333333333}{1000000000000000}}{{z}^{2}} + \frac{x \cdot \left(\left(\frac{91893853320467}{100000000000000} + \log x \cdot \left(x - \frac{1}{2}\right)\right) - x\right)}{{z}^{2}}\right)\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.7%

      \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right) - x}{z}, \frac{x}{z}, \frac{0.083333333333333}{z \cdot z}\right) + \left(0.0007936500793651 + y\right)\right) - \frac{0.0027777777777778}{z}\right) \cdot \left(z \cdot z\right)}{x} \]

   9. Taylor expanded in z around 0

      \[\leadsto z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \color{blue}{\left(\left(\left(\frac{91893853320467}{100000000000000} + \log x \cdot \left(x - \frac{1}{2}\right)\right) - x\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

   11. Applied rewrites 99.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, 0.0007936500793651 + y, \frac{-0.0027777777777778}{x}\right), \color{blue}{z}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - \left(x - \frac{0.083333333333333}{x}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, y + 0.0007936500793651, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - \left(x - \frac{0.083333333333333}{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{-1}{x}, \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, -6.298804484762296 \cdot 10^{-7}\right) \cdot \frac{z}{x}}{y - 0.0007936500793651} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -5e+134)
     (* (* (/ z x) y) z)
     (if (<= t_0 5e+307)
       (fma
        (-
         (fma
          (fma z 0.0007936500793651 -0.0027777777777778)
          z
          0.083333333333333))
        (/ -1.0 x)
        (fma (- x 0.5) (log x) (- 0.91893853320467 x)))
       (*
        (/
         (* (fma y y -6.298804484762296e-7) (/ z x))
         (- y 0.0007936500793651))
        z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -5e+134) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 5e+307) {
		tmp = fma(-fma(fma(z, 0.0007936500793651, -0.0027777777777778), z, 0.083333333333333), (-1.0 / x), fma((x - 0.5), log(x), (0.91893853320467 - x)));
	} else {
		tmp = ((fma(y, y, -6.298804484762296e-7) * (z / x)) / (y - 0.0007936500793651)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -5e+134)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_0 <= 5e+307)
		tmp = fma(Float64(-fma(fma(z, 0.0007936500793651, -0.0027777777777778), z, 0.083333333333333)), Float64(-1.0 / x), fma(Float64(x - 0.5), log(x), Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(Float64(fma(y, y, -6.298804484762296e-7) * Float64(z / x)) / Float64(y - 0.0007936500793651)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+134], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e+307], N[((-N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]) * N[(-1.0 / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y + -6.298804484762296e-7), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] / N[(y - 0.0007936500793651), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999981e134

   1. Initial program 84.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 87.5%

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

   if -4.99999999999999981e134 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 5e307

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      4. sub-neg N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot z + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} \cdot z + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. lower-fma.f64 94.8

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}, z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 94.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)\right)}{\mathsf{neg}\left(x\right)}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(x\right)}} + \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)\right), \frac{1}{\mathsf{neg}\left(x\right)}, \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} \]

   7. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{1}{-x}, \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\right)} \]

   if 5e307 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 86.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 92.2

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.3%

      \[\leadsto \frac{\mathsf{fma}\left(y, y, -6.298804484762296 \cdot 10^{-7}\right) \cdot \frac{z}{x}}{y - 0.0007936500793651} \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \leq -5 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), \frac{-1}{x}, \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, -6.298804484762296 \cdot 10^{-7}\right) \cdot \frac{z}{x}}{y - 0.0007936500793651} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, -6.298804484762296 \cdot 10^{-7}\right) \cdot \frac{z}{x}}{y - 0.0007936500793651} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x))))
   (if (<= t_0 -5e+134)
     (* (* (/ z x) y) z)
     (if (<= t_0 5e+307)
       (fma
        (- x 0.5)
        (log x)
        (+
         (/
          (fma
           (fma z 0.0007936500793651 -0.0027777777777778)
           z
           0.083333333333333)
          x)
         (- 0.91893853320467 x)))
       (*
        (/
         (* (fma y y -6.298804484762296e-7) (/ z x))
         (- y 0.0007936500793651))
        z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double tmp;
	if (t_0 <= -5e+134) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 5e+307) {
		tmp = fma((x - 0.5), log(x), ((fma(fma(z, 0.0007936500793651, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x)));
	} else {
		tmp = ((fma(y, y, -6.298804484762296e-7) * (z / x)) / (y - 0.0007936500793651)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
	tmp = 0.0
	if (t_0 <= -5e+134)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_0 <= 5e+307)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(fma(fma(z, 0.0007936500793651, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x)));
	else
		tmp = Float64(Float64(Float64(fma(y, y, -6.298804484762296e-7) * Float64(z / x)) / Float64(y - 0.0007936500793651)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+134], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e+307], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y + -6.298804484762296e-7), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] / N[(y - 0.0007936500793651), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -4.99999999999999981e134

   1. Initial program 84.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 87.5%

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

   if -4.99999999999999981e134 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 5e307

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}}{x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      4. sub-neg N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot z + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} \cdot z + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. lower-fma.f64 94.8

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}, z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 94.8%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}{x} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(x - \frac{1}{2}\right)} \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. lift-log.f64 N/A

         \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \color{blue}{\log x} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. sub-neg N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\mathsf{neg}\left(x\right)\right)\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x + \left(\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x}\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right)} \cdot \log x + \left(\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x}\right) \]

      11. lift-log.f64 N/A

          \[\leadsto \left(x - \frac{1}{2}\right) \cdot \color{blue}{\log x} + \left(\left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\left(\mathsf{neg}\left(x\right)\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right), z, \frac{83333333333333}{1000000000000000}\right)}{x}\right)} \]

   7. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) + \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\right)} \]

   if 5e307 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 86.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 92.2

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.3%

      \[\leadsto \frac{\mathsf{fma}\left(y, y, -6.298804484762296 \cdot 10^{-7}\right) \cdot \frac{z}{x}}{y - 0.0007936500793651} \cdot z \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \leq -5 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, -6.298804484762296 \cdot 10^{-7}\right) \cdot \frac{z}{x}}{y - 0.0007936500793651} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+84}:\\ \;\;\;\;\frac{y \cdot z}{x} \cdot z\\ \mathbf{elif}\;t\_0 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(y, x, 0.0007936500793651 \cdot x\right)}{x}}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -1e+84)
     (* (/ (* y z) x) z)
     (if (<= t_0 1e-9)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 0.083333333333333 x) 0.91893853320467) x))
       (if (<= t_0 5e+125)
         (/
          (fma
           (fma (+ y 0.0007936500793651) z -0.0027777777777778)
           z
           0.083333333333333)
          x)
         (* (* (/ (/ (fma y x (* 0.0007936500793651 x)) x) x) z) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -1e+84) {
		tmp = ((y * z) / x) * z;
	} else if (t_0 <= 1e-9) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	} else if (t_0 <= 5e+125) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((fma(y, x, (0.0007936500793651 * x)) / x) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -1e+84)
		tmp = Float64(Float64(Float64(y * z) / x) * z);
	elseif (t_0 <= 1e-9)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	elseif (t_0 <= 5e+125)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(fma(y, x, Float64(0.0007936500793651 * x)) / x) / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+84], N[(N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 1e-9], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+125], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(y * x + N[(0.0007936500793651 * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1.00000000000000006e84

   1. Initial program 85.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 79.0

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

   5. Applied rewrites 79.0%

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

   7. Applied rewrites 81.9%

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

   if -1.00000000000000006e84 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 1.00000000000000006e-9

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)} - x \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)\right)} - x \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{\log x \cdot \left(x - \frac{1}{2}\right) + \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} + \left(\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x - \frac{1}{2}}, \log x, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      7. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \color{blue}{\log x}, \left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x\right) \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right) - x}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{\left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \frac{91893853320467}{100000000000000}\right)} - x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\color{blue}{\frac{\frac{83333333333333}{1000000000000000} \cdot 1}{x}} + \frac{91893853320467}{100000000000000}\right) - x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} + \frac{91893853320467}{100000000000000}\right) - x\right) \]

      13. lower-/.f64 97.6

          \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(\color{blue}{\frac{0.083333333333333}{x}} + 0.91893853320467\right) - x\right) \]

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)} \]

   if 1.00000000000000006e-9 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 4.99999999999999962e125

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 75.5

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.0007936500793651 + y}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 4.99999999999999962e125 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 84.9

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.0%

      \[\leadsto \left(\frac{\frac{\mathsf{fma}\left(y, x, 0.0007936500793651 \cdot x\right)}{x}}{x} \cdot z\right) \cdot z \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -1 \cdot 10^{+84}:\\ \;\;\;\;\frac{y \cdot z}{x} \cdot z\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\mathsf{fma}\left(y, x, 0.0007936500793651 \cdot x\right)}{x}}{x} \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+37}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)}{x} - 1\right) \cdot x - \frac{\left(0.0027777777777778 - \left(y + 0.0007936500793651\right) \cdot z\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, y + 0.0007936500793651, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - \left(x - \frac{0.083333333333333}{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.4e+37)
   (-
    (* (- (/ (fma (log x) (- x 0.5) 0.91893853320467) x) 1.0) x)
    (/
     (-
      (* (- 0.0027777777777778 (* (+ y 0.0007936500793651) z)) z)
      0.083333333333333)
     x))
   (fma
    (fma (/ z x) (+ y 0.0007936500793651) (/ -0.0027777777777778 x))
    z
    (fma
     (log x)
     (- x 0.5)
     (- 0.91893853320467 (- x (/ 0.083333333333333 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.4e+37) {
		tmp = (((fma(log(x), (x - 0.5), 0.91893853320467) / x) - 1.0) * x) - ((((0.0027777777777778 - ((y + 0.0007936500793651) * z)) * z) - 0.083333333333333) / x);
	} else {
		tmp = fma(fma((z / x), (y + 0.0007936500793651), (-0.0027777777777778 / x)), z, fma(log(x), (x - 0.5), (0.91893853320467 - (x - (0.083333333333333 / x)))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.4e+37)
		tmp = Float64(Float64(Float64(Float64(fma(log(x), Float64(x - 0.5), 0.91893853320467) / x) - 1.0) * x) - Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(Float64(y + 0.0007936500793651) * z)) * z) - 0.083333333333333) / x));
	else
		tmp = fma(fma(Float64(z / x), Float64(y + 0.0007936500793651), Float64(-0.0027777777777778 / x)), z, fma(log(x), Float64(x - 0.5), Float64(0.91893853320467 - Float64(x - Float64(0.083333333333333 / x)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.4e+37], N[(N[(N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision] / x), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(N[(N[(0.0027777777777778 - N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision] + N[(-0.0027777777777778 / x), $MachinePrecision]), $MachinePrecision] * z + N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + N[(0.91893853320467 - N[(x - N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.40000000000000006e37

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{91893853320467}{100000000000000} \cdot \frac{1}{x}\right)\right) - 1\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{91893853320467}{100000000000000} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot x} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{91893853320467}{100000000000000} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot x} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\frac{-\log x}{x}, 0.5, \frac{0.91893853320467}{x}\right) + \log x\right) - 1\right) \cdot x} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \log x\right)}{x} - 1\right) \cdot x + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

      \[\leadsto \left(\frac{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)}{x} - 1\right) \cdot x + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 3.40000000000000006e37 < x

   1. Initial program 85.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   5. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \frac{{z}^{2} \cdot \left(\left(\frac{7936500793651}{10000000000000000} + \left(y + \left(\frac{\frac{83333333333333}{1000000000000000}}{{z}^{2}} + \frac{x \cdot \left(\left(\frac{91893853320467}{100000000000000} + \log x \cdot \left(x - \frac{1}{2}\right)\right) - x\right)}{{z}^{2}}\right)\right)\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{z}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.1%

      \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right) - x}{z}, \frac{x}{z}, \frac{0.083333333333333}{z \cdot z}\right) + \left(0.0007936500793651 + y\right)\right) - \frac{0.0027777777777778}{z}\right) \cdot \left(z \cdot z\right)}{x} \]

   9. Taylor expanded in z around 0

      \[\leadsto z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \color{blue}{\left(\left(\left(\frac{91893853320467}{100000000000000} + \log x \cdot \left(x - \frac{1}{2}\right)\right) - x\right) + \frac{83333333333333}{1000000000000000} \cdot \frac{1}{x}\right)} \]

   11. Applied rewrites 99.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, 0.0007936500793651 + y, \frac{-0.0027777777777778}{x}\right), \color{blue}{z}, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - \left(x - \frac{0.083333333333333}{x}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{+37}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)}{x} - 1\right) \cdot x - \frac{\left(0.0027777777777778 - \left(y + 0.0007936500793651\right) \cdot z\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{z}{x}, y + 0.0007936500793651, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467 - \left(x - \frac{0.083333333333333}{x}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{t\_0 + 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y, -6.298804484762296 \cdot 10^{-7}\right) \cdot \frac{z}{x}}{y - 0.0007936500793651} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 5e+307)
     (+
      (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
      (/ (+ t_0 0.083333333333333) x))
     (*
      (/ (* (fma y y -6.298804484762296e-7) (/ z x)) (- y 0.0007936500793651))
      z))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= 5e+307) {
		tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((t_0 + 0.083333333333333) / x);
	} else {
		tmp = ((fma(y, y, -6.298804484762296e-7) * (z / x)) / (y - 0.0007936500793651)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= 5e+307)
		tmp = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(t_0 + 0.083333333333333) / x));
	else
		tmp = Float64(Float64(Float64(fma(y, y, -6.298804484762296e-7) * Float64(z / x)) / Float64(y - 0.0007936500793651)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+307], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(t$95$0 + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y + -6.298804484762296e-7), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision] / N[(y - 0.0007936500793651), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5e307

   1. Initial program 97.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   if 5e307 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 82.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 91.9

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, y, -6.298804484762296 \cdot 10^{-7}\right) \cdot \frac{z}{x}}{y - 0.0007936500793651} \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+96}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)}{x} - 1\right) \cdot x - \frac{\left(0.0027777777777778 - \left(y + 0.0007936500793651\right) \cdot z\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.15e+96)
   (-
    (* (- (/ (fma (log x) (- x 0.5) 0.91893853320467) x) 1.0) x)
    (/
     (-
      (* (- 0.0027777777777778 (* (+ y 0.0007936500793651) z)) z)
      0.083333333333333)
     x))
   (fma
    (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
    z
    (-
     (+ (fma (- x 0.5) (log x) 0.91893853320467) (/ 0.083333333333333 x))
     x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.15e+96) {
		tmp = (((fma(log(x), (x - 0.5), 0.91893853320467) / x) - 1.0) * x) - ((((0.0027777777777778 - ((y + 0.0007936500793651) * z)) * z) - 0.083333333333333) / x);
	} else {
		tmp = fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, ((fma((x - 0.5), log(x), 0.91893853320467) + (0.083333333333333 / x)) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.15e+96)
		tmp = Float64(Float64(Float64(Float64(fma(log(x), Float64(x - 0.5), 0.91893853320467) / x) - 1.0) * x) - Float64(Float64(Float64(Float64(0.0027777777777778 - Float64(Float64(y + 0.0007936500793651) * z)) * z) - 0.083333333333333) / x));
	else
		tmp = fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, Float64(Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) + Float64(0.083333333333333 / x)) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.15e+96], N[(N[(N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision] / x), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(N[(N[(0.0027777777777778 - N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15000000000000008e96

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{91893853320467}{100000000000000} \cdot \frac{1}{x}\right)\right) - 1\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{91893853320467}{100000000000000} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot x} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(\frac{1}{2} \cdot \frac{\log \left(\frac{1}{x}\right)}{x} + \frac{91893853320467}{100000000000000} \cdot \frac{1}{x}\right)\right) - 1\right) \cdot x} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\frac{-\log x}{x}, 0.5, \frac{0.91893853320467}{x}\right) + \log x\right) - 1\right) \cdot x} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \log x\right)}{x} - 1\right) \cdot x + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.7%

      \[\leadsto \left(\frac{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)}{x} - 1\right) \cdot x + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 1.15000000000000008e96 < x

   1. Initial program 81.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 16.1

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

   5. Applied rewrites 16.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]

   11. Applied rewrites 90.4%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)\right) - x} \]
   12. Step-by-step derivation

   13. Applied rewrites 90.4%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, \color{blue}{z}, \left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \frac{0.083333333333333}{x}\right) - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+96}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)}{x} - 1\right) \cdot x - \frac{\left(0.0027777777777778 - \left(y + 0.0007936500793651\right) \cdot z\right) \cdot z - 0.083333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8e+104)
   (/
    (fma
     (+ (fma (- x 0.5) (log x) (- x)) 0.91893853320467)
     x
     (fma
      (fma (+ y 0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333))
    x)
   (fma
    (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
    z
    (-
     (+ (fma (- x 0.5) (log x) 0.91893853320467) (/ 0.083333333333333 x))
     x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8e+104) {
		tmp = fma((fma((x - 0.5), log(x), -x) + 0.91893853320467), x, fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)) / x;
	} else {
		tmp = fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, ((fma((x - 0.5), log(x), 0.91893853320467) + (0.083333333333333 / x)) - x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8e+104)
		tmp = Float64(fma(Float64(fma(Float64(x - 0.5), log(x), Float64(-x)) + 0.91893853320467), x, fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)) / x);
	else
		tmp = fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, Float64(Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) + Float64(0.083333333333333 / x)) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8e+104], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x + N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e104

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

   if 8e104 < x

   1. Initial program 81.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 15.2

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

   5. Applied rewrites 15.2%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]

   11. Applied rewrites 90.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)\right) - x} \]
   12. Step-by-step derivation

   13. Applied rewrites 90.2%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, \color{blue}{z}, \left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \frac{0.083333333333333}{x}\right) - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \left(\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \frac{0.083333333333333}{x}\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, 0.91893853320467\right) + \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8e+104)
   (/
    (fma
     (+ (fma (- x 0.5) (log x) (- x)) 0.91893853320467)
     x
     (fma
      (fma (+ y 0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333))
    x)
   (-
    (+
     (fma
      (/ (fma 0.0007936500793651 z -0.0027777777777778) x)
      z
      0.91893853320467)
     (fma (- x 0.5) (log x) (/ 0.083333333333333 x)))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8e+104) {
		tmp = fma((fma((x - 0.5), log(x), -x) + 0.91893853320467), x, fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)) / x;
	} else {
		tmp = (fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, 0.91893853320467) + fma((x - 0.5), log(x), (0.083333333333333 / x))) - x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8e+104)
		tmp = Float64(fma(Float64(fma(Float64(x - 0.5), log(x), Float64(-x)) + 0.91893853320467), x, fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)) / x);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, 0.91893853320467) + fma(Float64(x - 0.5), log(x), Float64(0.083333333333333 / x))) - x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8e+104], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x + N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + 0.91893853320467), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e104

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

   if 8e104 < x

   1. Initial program 81.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]

   5. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, 0.91893853320467\right)\right) - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, 0.91893853320467\right) + \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.7e+159)
   (/
    (fma
     (+ (fma (- x 0.5) (log x) (- x)) 0.91893853320467)
     x
     (fma
      (fma (+ y 0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333))
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.7e+159) {
		tmp = fma((fma((x - 0.5), log(x), -x) + 0.91893853320467), x, fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.7e+159)
		tmp = Float64(fma(Float64(fma(Float64(x - 0.5), log(x), Float64(-x)) + 0.91893853320467), x, fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.7e+159], N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x + N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996e159

   1. Initial program 97.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

   if 1.69999999999999996e159 < x

   1. Initial program 82.7%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 14.8

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

   5. Applied rewrites 14.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. mul-1-neg N/A

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

       5. log-rec N/A

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

       6. remove-double-neg N/A

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

       7. lower-log.f64 82.8

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

   11. Applied rewrites 82.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x - 1\\ \mathbf{if}\;x \leq 1.08 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+86}:\\ \;\;\;\;\left(\frac{t\_0}{y} \cdot x\right) \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+113}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log x) 1.0)))
   (if (<= x 1.08e+17)
     (/
      (fma
       (fma (+ y 0.0007936500793651) z -0.0027777777777778)
       z
       0.083333333333333)
      x)
     (if (<= x 8.8e+86)
       (* (* (/ t_0 y) x) y)
       (if (<= x 5.5e+113)
         (* (* (+ (/ 0.0007936500793651 x) (/ y x)) z) z)
         (* t_0 x))))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) - 1.0;
	double tmp;
	if (x <= 1.08e+17) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else if (x <= 8.8e+86) {
		tmp = ((t_0 / y) * x) * y;
	} else if (x <= 5.5e+113) {
		tmp = (((0.0007936500793651 / x) + (y / x)) * z) * z;
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(x) - 1.0)
	tmp = 0.0
	if (x <= 1.08e+17)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	elseif (x <= 8.8e+86)
		tmp = Float64(Float64(Float64(t_0 / y) * x) * y);
	elseif (x <= 5.5e+113)
		tmp = Float64(Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * z) * z);
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, 1.08e+17], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 8.8e+86], N[(N[(N[(t$95$0 / y), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 5.5e+113], N[(N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.08e17

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 94.2

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.0007936500793651 + y}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 1.08e17 < x < 8.80000000000000013e86

   1. Initial program 99.4%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x}{y} + -1 \cdot \frac{{z}^{2}}{x}\right)\right)} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{z}{x}, \frac{\left(\mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, 0.91893853320467\right)\right) - x}{y}\right) \cdot y} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.3%

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

   if 8.80000000000000013e86 < x < 5.5000000000000001e113

   1. Initial program 76.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 83.5

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]

   if 5.5000000000000001e113 < x

   1. Initial program 83.5%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 15.0

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

   5. Applied rewrites 15.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. mul-1-neg N/A

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

       5. log-rec N/A

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

       6. remove-double-neg N/A

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

       7. lower-log.f64 81.0

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

   11. Applied rewrites 81.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.08 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+86}:\\ \;\;\;\;\left(\frac{\log x - 1}{y} \cdot x\right) \cdot y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+113}:\\ \;\;\;\;\left(\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.08 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.08e+17)
   (/
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.08e+17) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.08e+17)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.08e+17], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.08e17

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 94.2

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.0007936500793651 + y}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 1.08e17 < x

   1. Initial program 86.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 19.1

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

   5. Applied rewrites 19.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. mul-1-neg N/A

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

       5. log-rec N/A

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

       6. remove-double-neg N/A

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

       7. lower-log.f64 71.0

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

   11. Applied rewrites 71.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.08 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := \frac{z}{x} \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 0.0007936500793651\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z))
        (t_1 (* (/ z x) z)))
   (if (<= t_0 -2e+58)
     (* (* (/ z x) y) z)
     (if (<= t_0 5e+77)
       (/
        (fma
         (fma 0.0007936500793651 z -0.0027777777777778)
         z
         0.083333333333333)
        x)
       (if (<= t_0 5e+140) (* t_1 y) (* t_1 0.0007936500793651))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = (z / x) * z;
	double tmp;
	if (t_0 <= -2e+58) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 5e+77) {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else if (t_0 <= 5e+140) {
		tmp = t_1 * y;
	} else {
		tmp = t_1 * 0.0007936500793651;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	t_1 = Float64(Float64(z / x) * z)
	tmp = 0.0
	if (t_0 <= -2e+58)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_0 <= 5e+77)
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	elseif (t_0 <= 5e+140)
		tmp = Float64(t_1 * y);
	else
		tmp = Float64(t_1 * 0.0007936500793651);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+58], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 5e+77], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e+140], N[(t$95$1 * y), $MachinePrecision], N[(t$95$1 * 0.0007936500793651), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1.99999999999999989e58

   1. Initial program 85.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 80.1%

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

   if -1.99999999999999989e58 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000004e77

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]

   11. Applied rewrites 98.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)\right) - x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]
   13. Step-by-step derivation

   14. Applied rewrites 47.2%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{\color{blue}{x}} \]

   if 5.00000000000000004e77 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.00000000000000008e140

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x}{y} + -1 \cdot \frac{{z}^{2}}{x}\right)\right)} \]

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{z}{x}, \frac{\left(\mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, 0.91893853320467\right)\right) - x}{y}\right) \cdot y} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{{z}^{2}}{x} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 60.5%

      \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot y \]

   if 5.00000000000000008e140 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 87.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 86.1

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.6%

      \[\leadsto \left(\frac{z}{x} \cdot z\right) \cdot \color{blue}{0.0007936500793651} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)))
   (if (<= t_0 -2e+58)
     (* (* (/ z x) y) z)
     (if (<= t_0 2e+64)
       (/
        (fma
         (fma 0.0007936500793651 z -0.0027777777777778)
         z
         0.083333333333333)
        x)
       (* (* (/ z x) (+ y 0.0007936500793651)) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double tmp;
	if (t_0 <= -2e+58) {
		tmp = ((z / x) * y) * z;
	} else if (t_0 <= 2e+64) {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * (y + 0.0007936500793651)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_0 <= -2e+58)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_0 <= 2e+64)
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * Float64(y + 0.0007936500793651)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+58], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$0, 2e+64], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1.99999999999999989e58

   1. Initial program 85.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 80.1%

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

   if -1.99999999999999989e58 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.00000000000000004e64

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]

   11. Applied rewrites 99.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)\right) - x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]
   13. Step-by-step derivation

   14. Applied rewrites 46.5%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{\color{blue}{x}} \]

   if 2.00000000000000004e64 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

       2. unpow2 N/A

          \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

       6. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z + \frac{y}{x} \cdot z\right)} \cdot z \]

       7. associate-*r/ N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}} \cdot z + \frac{y}{x} \cdot z\right) \cdot z \]

       8. metadata-eval N/A

          \[\leadsto \left(\frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} \cdot z + \frac{y}{x} \cdot z\right) \cdot z \]

       9. associate-*l/ N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot z}{x}} + \frac{y}{x} \cdot z\right) \cdot z \]

       10. associate-*r/ N/A

           \[\leadsto \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}} + \frac{y}{x} \cdot z\right) \cdot z \]

       11. associate-*l/ N/A

           \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{\frac{y \cdot z}{x}}\right) \cdot z \]

       12. associate-/l* N/A

           \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x} + \color{blue}{y \cdot \frac{z}{x}}\right) \cdot z \]

       13. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \cdot z \]

       14. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(\frac{z}{x} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)} \cdot z \]

       15. lower-/.f64 N/A

           \[\leadsto \left(\color{blue}{\frac{z}{x}} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot z \]

       16. lower-+.f64 79.6

           \[\leadsto \left(\frac{z}{x} \cdot \color{blue}{\left(0.0007936500793651 + y\right)}\right) \cdot z \]

   11. Applied rewrites 79.6%

       \[\leadsto \color{blue}{\left(\frac{z}{x} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot \left(y + 0.0007936500793651\right)\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + 0.0007936500793651\right) \cdot z\\ t_1 := \left(t\_0 - 0.0027777777777778\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y 0.0007936500793651) z))
        (t_1 (* (- t_0 0.0027777777777778) z)))
   (if (<= t_1 -2e+58)
     (* (* (/ z x) y) z)
     (if (<= t_1 2e+25)
       (/
        (fma
         (fma 0.0007936500793651 z -0.0027777777777778)
         z
         0.083333333333333)
        x)
       (/ (* t_0 z) x)))))

C

double code(double x, double y, double z) {
	double t_0 = (y + 0.0007936500793651) * z;
	double t_1 = (t_0 - 0.0027777777777778) * z;
	double tmp;
	if (t_1 <= -2e+58) {
		tmp = ((z / x) * y) * z;
	} else if (t_1 <= 2e+25) {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (t_0 * z) / x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + 0.0007936500793651) * z)
	t_1 = Float64(Float64(t_0 - 0.0027777777777778) * z)
	tmp = 0.0
	if (t_1 <= -2e+58)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (t_1 <= 2e+25)
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(t_0 * z) / x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+58], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, 2e+25], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(t$95$0 * z), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < -1.99999999999999989e58

   1. Initial program 85.8%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 80.1%

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

   if -1.99999999999999989e58 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 2.00000000000000018e25

   1. Initial program 99.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 2.4

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

   5. Applied rewrites 2.4%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) - \frac{13888888888889}{5000000000000000} \cdot \frac{1}{x}\right) + \log x \cdot \left(x - \frac{1}{2}\right)\right)\right)\right) - x} \]

   8. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x} + \frac{0.0007936500793651}{x}, z, \frac{-0.0027777777777778}{x}\right), z, \mathsf{fma}\left(\log x, x - 0.5, \frac{0.083333333333333}{x}\right)\right) + \left(0.91893853320467 - x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x} \]

   11. Applied rewrites 99.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)\right) - x} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]
   13. Step-by-step derivation

   14. Applied rewrites 45.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{\color{blue}{x}} \]

   if 2.00000000000000018e25 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 90.2%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + \left(x \cdot \left(\frac{91893853320467}{100000000000000} + \left(\frac{-1}{2} \cdot \log x + x \cdot \left(\log x - 1\right)\right)\right) + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)\right)}{x}} \]

   5. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x - 0.5, \log x, -x\right) + 0.91893853320467, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}} \]

   6. Taylor expanded in z around inf

      \[\leadsto \frac{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.0%

      \[\leadsto \frac{\left(\left(0.0007936500793651 + y\right) \cdot z\right) \cdot z}{\color{blue}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z\right) \cdot z}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -0.01:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.00079365008:\\ \;\;\;\;\left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ y 0.0007936500793651) -0.01)
   (* (* (/ z x) y) z)
   (if (<= (+ y 0.0007936500793651) 0.00079365008)
     (* (* (/ z x) 0.0007936500793651) z)
     (* (* (/ y x) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -0.01) {
		tmp = ((z / x) * y) * z;
	} else if ((y + 0.0007936500793651) <= 0.00079365008) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = ((y / 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 ((y + 0.0007936500793651d0) <= (-0.01d0)) then
        tmp = ((z / x) * y) * z
    else if ((y + 0.0007936500793651d0) <= 0.00079365008d0) then
        tmp = ((z / x) * 0.0007936500793651d0) * z
    else
        tmp = ((y / x) * z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -0.01) {
		tmp = ((z / x) * y) * z;
	} else if ((y + 0.0007936500793651) <= 0.00079365008) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = ((y / x) * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y + 0.0007936500793651) <= -0.01:
		tmp = ((z / x) * y) * z
	elif (y + 0.0007936500793651) <= 0.00079365008:
		tmp = ((z / x) * 0.0007936500793651) * z
	else:
		tmp = ((y / x) * z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -0.01)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (Float64(y + 0.0007936500793651) <= 0.00079365008)
		tmp = Float64(Float64(Float64(z / x) * 0.0007936500793651) * z);
	else
		tmp = Float64(Float64(Float64(y / x) * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y + 0.0007936500793651) <= -0.01)
		tmp = ((z / x) * y) * z;
	elseif ((y + 0.0007936500793651) <= 0.00079365008)
		tmp = ((z / x) * 0.0007936500793651) * z;
	else
		tmp = ((y / x) * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -0.01], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 0.00079365008], N[(N[(N[(z / x), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -0.0100000000000000002

   1. Initial program 91.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 46.6

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 48.0%

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

   9. Applied rewrites 48.1%

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

   if -0.0100000000000000002 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 7.9365008000000003e-4

   1. Initial program 93.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 40.5

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 40.5%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 40.5%

      \[\leadsto \left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z \]

   if 7.9365008000000003e-4 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 96.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 54.7%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -0.01:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.00079365008:\\ \;\;\;\;\left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -0.01:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.00079365008:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ y 0.0007936500793651) -0.01)
   (* (* (/ z x) y) z)
   (if (<= (+ y 0.0007936500793651) 0.00079365008)
     (* (* (/ z x) z) 0.0007936500793651)
     (* (* (/ y x) z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -0.01) {
		tmp = ((z / x) * y) * z;
	} else if ((y + 0.0007936500793651) <= 0.00079365008) {
		tmp = ((z / x) * z) * 0.0007936500793651;
	} else {
		tmp = ((y / 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 ((y + 0.0007936500793651d0) <= (-0.01d0)) then
        tmp = ((z / x) * y) * z
    else if ((y + 0.0007936500793651d0) <= 0.00079365008d0) then
        tmp = ((z / x) * z) * 0.0007936500793651d0
    else
        tmp = ((y / x) * z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y + 0.0007936500793651) <= -0.01) {
		tmp = ((z / x) * y) * z;
	} else if ((y + 0.0007936500793651) <= 0.00079365008) {
		tmp = ((z / x) * z) * 0.0007936500793651;
	} else {
		tmp = ((y / x) * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y + 0.0007936500793651) <= -0.01:
		tmp = ((z / x) * y) * z
	elif (y + 0.0007936500793651) <= 0.00079365008:
		tmp = ((z / x) * z) * 0.0007936500793651
	else:
		tmp = ((y / x) * z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -0.01)
		tmp = Float64(Float64(Float64(z / x) * y) * z);
	elseif (Float64(y + 0.0007936500793651) <= 0.00079365008)
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	else
		tmp = Float64(Float64(Float64(y / x) * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y + 0.0007936500793651) <= -0.01)
		tmp = ((z / x) * y) * z;
	elseif ((y + 0.0007936500793651) <= 0.00079365008)
		tmp = ((z / x) * z) * 0.0007936500793651;
	else
		tmp = ((y / x) * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -0.01], N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 0.00079365008], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -0.0100000000000000002

   1. Initial program 91.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 46.6

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

   5. Applied rewrites 46.6%

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

   7. Applied rewrites 48.0%

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

   9. Applied rewrites 48.1%

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

   if -0.0100000000000000002 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 7.9365008000000003e-4

   1. Initial program 93.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 40.5

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 40.5%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.3%

      \[\leadsto \left(\frac{z}{x} \cdot z\right) \cdot \color{blue}{0.0007936500793651} \]

   if 7.9365008000000003e-4 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 96.9%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 54.7%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -0.01:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.00079365008:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x} \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{if}\;y + 0.0007936500793651 \leq -0.01:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.00079365008:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ z x) y) z)))
   (if (<= (+ y 0.0007936500793651) -0.01)
     t_0
     (if (<= (+ y 0.0007936500793651) 0.00079365008)
       (* (* (/ z x) z) 0.0007936500793651)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z / x) * y) * z;
	double tmp;
	if ((y + 0.0007936500793651) <= -0.01) {
		tmp = t_0;
	} else if ((y + 0.0007936500793651) <= 0.00079365008) {
		tmp = ((z / x) * z) * 0.0007936500793651;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((z / x) * y) * z
    if ((y + 0.0007936500793651d0) <= (-0.01d0)) then
        tmp = t_0
    else if ((y + 0.0007936500793651d0) <= 0.00079365008d0) then
        tmp = ((z / x) * z) * 0.0007936500793651d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((z / x) * y) * z;
	double tmp;
	if ((y + 0.0007936500793651) <= -0.01) {
		tmp = t_0;
	} else if ((y + 0.0007936500793651) <= 0.00079365008) {
		tmp = ((z / x) * z) * 0.0007936500793651;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((z / x) * y) * z
	tmp = 0
	if (y + 0.0007936500793651) <= -0.01:
		tmp = t_0
	elif (y + 0.0007936500793651) <= 0.00079365008:
		tmp = ((z / x) * z) * 0.0007936500793651
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z / x) * y) * z)
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -0.01)
		tmp = t_0;
	elseif (Float64(y + 0.0007936500793651) <= 0.00079365008)
		tmp = Float64(Float64(Float64(z / x) * z) * 0.0007936500793651);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((z / x) * y) * z;
	tmp = 0.0;
	if ((y + 0.0007936500793651) <= -0.01)
		tmp = t_0;
	elseif ((y + 0.0007936500793651) <= 0.00079365008)
		tmp = ((z / x) * z) * 0.0007936500793651;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z / x), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -0.01], t$95$0, If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 0.00079365008], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -0.0100000000000000002 or 7.9365008000000003e-4 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 94.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{z}^{2} \cdot y}}{x} \]

      4. unpow2 N/A

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

      5. lower-*.f64 53.0

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 51.6%

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

   9. Applied rewrites 51.6%

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

   if -0.0100000000000000002 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 7.9365008000000003e-4

   1. Initial program 93.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 40.5

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 40.5%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.3%

      \[\leadsto \left(\frac{z}{x} \cdot z\right) \cdot \color{blue}{0.0007936500793651} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + 0.0007936500793651 \leq -0.01:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.00079365008:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{if}\;y \leq -0.0008:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ z x) z) y)))
   (if (<= y -0.0008)
     t_0
     (if (<= y 3.4e-12) (* (* (/ z x) 0.0007936500793651) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z / x) * z) * y;
	double tmp;
	if (y <= -0.0008) {
		tmp = t_0;
	} else if (y <= 3.4e-12) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((z / x) * z) * y
    if (y <= (-0.0008d0)) then
        tmp = t_0
    else if (y <= 3.4d-12) then
        tmp = ((z / x) * 0.0007936500793651d0) * z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((z / x) * z) * y;
	double tmp;
	if (y <= -0.0008) {
		tmp = t_0;
	} else if (y <= 3.4e-12) {
		tmp = ((z / x) * 0.0007936500793651) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((z / x) * z) * y
	tmp = 0
	if y <= -0.0008:
		tmp = t_0
	elif y <= 3.4e-12:
		tmp = ((z / x) * 0.0007936500793651) * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z / x) * z) * y)
	tmp = 0.0
	if (y <= -0.0008)
		tmp = t_0;
	elseif (y <= 3.4e-12)
		tmp = Float64(Float64(Float64(z / x) * 0.0007936500793651) * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((z / x) * z) * y;
	tmp = 0.0;
	if (y <= -0.0008)
		tmp = t_0;
	elseif (y <= 3.4e-12)
		tmp = ((z / x) * 0.0007936500793651) * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -0.0008], t$95$0, If[LessEqual[y, 3.4e-12], N[(N[(N[(z / x), $MachinePrecision] * 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000038e-4 or 3.4000000000000001e-12 < y

   1. Initial program 94.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(-1 \cdot \frac{\left(\frac{91893853320467}{100000000000000} + \left(\frac{83333333333333}{1000000000000000} \cdot \frac{1}{x} + \left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot z - \frac{13888888888889}{5000000000000000}\right)}{x}\right)\right)\right) - x}{y} + -1 \cdot \frac{{z}^{2}}{x}\right)\right)} \]

   5. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{z}{x}, \frac{\left(\mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right) + \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, 0.91893853320467\right)\right) - x}{y}\right) \cdot y} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{{z}^{2}}{x} \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 54.4%

      \[\leadsto \left(z \cdot \frac{z}{x}\right) \cdot y \]

   if -8.00000000000000038e-4 < y < 3.4000000000000001e-12

   1. Initial program 93.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 40.5

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 40.5%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{z}{x}\right) \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 40.5%

      \[\leadsto \left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0008:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{z}{x} \cdot 0.0007936500793651\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.26 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + 0.0007936500793651\right) \cdot z}{x} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.26e+104)
   (/
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (/ (* (+ y 0.0007936500793651) z) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.26e+104) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((y + 0.0007936500793651) * z) / x) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.26e+104)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) / x) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.26e+104], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25999999999999994e104

   1. Initial program 99.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}}{x} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z} + \frac{83333333333333}{1000000000000000}}{x} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z} + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{7936500793651}{10000000000000000} + y\right) \cdot z + \color{blue}{\frac{-13888888888889}{5000000000000000}}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000} + y, z, \frac{-13888888888889}{5000000000000000}\right)}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]

      9. lower-+.f64 85.0

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.0007936500793651 + y}, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   5. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}} \]

   if 1.25999999999999994e104 < x

   1. Initial program 81.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

      8. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      9. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

      11. associate-*r/ N/A

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

      12. metadata-eval N/A

          \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

      13. lower-/.f64 25.9

          \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

   5. Applied rewrites 25.9%

      \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 25.9%

      \[\leadsto \frac{\left(y + 0.0007936500793651\right) \cdot z}{x} \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.26 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + 0.0007936500793651\right) \cdot z}{x} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \left(\frac{z}{x} \cdot z\right) \cdot 0.0007936500793651 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* (/ z x) z) 0.0007936500793651))

C

double code(double x, double y, double z) {
	return ((z / x) * z) * 0.0007936500793651;
}

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 / x) * z) * 0.0007936500793651d0
end function

Java

public static double code(double x, double y, double z) {
	return ((z / x) * z) * 0.0007936500793651;
}

Python

def code(x, y, z):
	return ((z / x) * z) * 0.0007936500793651

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z / x) * z) * 0.0007936500793651)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z / x) * z) * 0.0007936500793651;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * 0.0007936500793651), $MachinePrecision]

Derivation

1. Initial program 93.7%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot {z}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot \color{blue}{\left(z \cdot z\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right) \cdot z} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right)} \cdot z \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)\right) \cdot z} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right) \cdot z\right)} \cdot z \]

   8. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\frac{y}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right)} \cdot z\right) \cdot z \]

   10. lower-/.f64 N/A

       \[\leadsto \left(\left(\color{blue}{\frac{y}{x}} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}\right) \cdot z\right) \cdot z \]

   11. associate-*r/ N/A

       \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot 1}{x}}\right) \cdot z\right) \cdot z \]

   12. metadata-eval N/A

       \[\leadsto \left(\left(\frac{y}{x} + \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x}\right) \cdot z\right) \cdot z \]

   13. lower-/.f64 46.0

       \[\leadsto \left(\left(\frac{y}{x} + \color{blue}{\frac{0.0007936500793651}{x}}\right) \cdot z\right) \cdot z \]

5. Applied rewrites 46.0%

   \[\leadsto \color{blue}{\left(\left(\frac{y}{x} + \frac{0.0007936500793651}{x}\right) \cdot z\right) \cdot z} \]

6. Taylor expanded in y around 0

   \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \color{blue}{\frac{{z}^{2}}{x}} \]
7. Step-by-step derivation

8. Applied rewrites 29.5%

   \[\leadsto \left(\frac{z}{x} \cdot z\right) \cdot \color{blue}{0.0007936500793651} \]
9. Add Preprocessing

Developer Target 1: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467 - x\right)\right) + \frac{0.083333333333333}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 0.0007936500793651\right) - 0.0027777777777778\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x))
  (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778))))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

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 - 0.5d0) * log(x)) + (0.91893853320467d0 - x)) + (0.083333333333333d0 / x)) + ((z / x) * ((z * (y + 0.0007936500793651d0)) - 0.0027777777777778d0))
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) + Float64(0.91893853320467 - x)) + Float64(0.083333333333333 / x)) + Float64(Float64(z / x) * Float64(Float64(z * Float64(y + 0.0007936500793651)) - 0.0027777777777778)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) + (0.91893853320467 - x)) + (0.083333333333333 / x)) + ((z / x) * ((z * (y + 0.0007936500793651)) - 0.0027777777777778));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / x), $MachinePrecision] * N[(N[(z * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024285 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (* (- x 1/2) (log x)) (- 91893853320467/100000000000000 x)) (/ 83333333333333/1000000000000000 x)) (* (/ z x) (- (* z (+ y 7936500793651/10000000000000000)) 13888888888889/5000000000000000))))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))