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

Percentage Accurate: 93.7% → 99.4%

Time: 15.1s

Alternatives: 19

Speedup: 0.9×

• 31.0% 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: 93.7% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+56)
   (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 (/ y x) z (/ (fma 0.0007936500793651 z -0.0027777777777778) x))
     z
     (fma (- x 0.5) (log x) (/ 0.083333333333333 x)))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+56) {
		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((y / x), z, (fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma((x - 0.5), log(x), (0.083333333333333 / x))) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+56)
		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 = Float64(fma(fma(Float64(y / x), z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma(Float64(x - 0.5), log(x), Float64(0.083333333333333 / x))) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e+56], 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[(N[(y / x), $MachinePrecision] * z + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000009e56

   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. 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 1.00000000000000009e56 < x

   1. Initial program 87.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y + 0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right), {x}^{-1}, \mathsf{fma}\left(\log x, x - 0.5, \mathsf{fma}\left(-1, x, 0.91893853320467\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{\frac{0.0007936500793651}{x}}{y}\right), y, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + 0.0007936500793651\right) \cdot {x}^{-1}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))
   (if (<= t_0 -2e+82)
     (fma
      (* (* z z) (+ (/ 1.0 x) (/ (/ 0.0007936500793651 x) y)))
      y
      (/ 0.083333333333333 x))
     (if (<= t_0 2e+299)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 0.083333333333333 x) 0.91893853320467) x))
       (* (* (* (+ y 0.0007936500793651) (pow x -1.0)) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	double tmp;
	if (t_0 <= -2e+82) {
		tmp = fma(((z * z) * ((1.0 / x) + ((0.0007936500793651 / x) / y))), y, (0.083333333333333 / x));
	} else if (t_0 <= 2e+299) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	} else {
		tmp = (((y + 0.0007936500793651) * pow(x, -1.0)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467))
	tmp = 0.0
	if (t_0 <= -2e+82)
		tmp = fma(Float64(Float64(z * z) * Float64(Float64(1.0 / x) + Float64(Float64(0.0007936500793651 / x) / y))), y, Float64(0.083333333333333 / x));
	elseif (t_0 <= 2e+299)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	else
		tmp = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * (x ^ -1.0)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+82], N[(N[(N[(z * z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.0007936500793651 / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+299], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(N[(0.083333333333333 / x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] * z), $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)) < -1.9999999999999999e82

   1. Initial program 90.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left({z}^{2} \cdot \left(\frac{1}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right), y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.4%

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

   if -1.9999999999999999e82 < (+.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)) < 2.0000000000000001e299

   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 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 86.7

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

   5. Applied rewrites 86.7%

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

   if 2.0000000000000001e299 < (+.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 87.0%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 90.7%

      \[\leadsto \left(\left({x}^{-1} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) \cdot \color{blue}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{\frac{0.0007936500793651}{x}}{y}\right), y, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + 0.0007936500793651\right) \cdot {x}^{-1}\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{\frac{0.0007936500793651}{x}}{y}\right), y, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(\left(\log x \cdot x - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + 0.0007936500793651\right) \cdot {x}^{-1}\right) \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))
   (if (<= t_0 -2e+82)
     (fma
      (* (* z z) (+ (/ 1.0 x) (/ (/ 0.0007936500793651 x) y)))
      y
      (/ 0.083333333333333 x))
     (if (<= t_0 2e+299)
       (+ (/ 0.083333333333333 x) (+ (- (* (log x) x) x) 0.91893853320467))
       (* (* (* (+ y 0.0007936500793651) (pow x -1.0)) z) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	double tmp;
	if (t_0 <= -2e+82) {
		tmp = fma(((z * z) * ((1.0 / x) + ((0.0007936500793651 / x) / y))), y, (0.083333333333333 / x));
	} else if (t_0 <= 2e+299) {
		tmp = (0.083333333333333 / x) + (((log(x) * x) - x) + 0.91893853320467);
	} else {
		tmp = (((y + 0.0007936500793651) * pow(x, -1.0)) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467))
	tmp = 0.0
	if (t_0 <= -2e+82)
		tmp = fma(Float64(Float64(z * z) * Float64(Float64(1.0 / x) + Float64(Float64(0.0007936500793651 / x) / y))), y, Float64(0.083333333333333 / x));
	elseif (t_0 <= 2e+299)
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(Float64(Float64(log(x) * x) - x) + 0.91893853320467));
	else
		tmp = Float64(Float64(Float64(Float64(y + 0.0007936500793651) * (x ^ -1.0)) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+82], N[(N[(N[(z * z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.0007936500793651 / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+299], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision] * z), $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)) < -1.9999999999999999e82

   1. Initial program 90.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left({z}^{2} \cdot \left(\frac{1}{x} + \frac{7936500793651}{10000000000000000} \cdot \frac{1}{x \cdot y}\right), y, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.4%

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

   if -1.9999999999999999e82 < (+.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)) < 2.0000000000000001e299

   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 z around 0

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 84.8

         \[\leadsto \left(\left(\color{blue}{\log x} \cdot x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

   8. Applied rewrites 84.8%

      \[\leadsto \left(\left(\color{blue}{\log x \cdot x} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

   if 2.0000000000000001e299 < (+.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 87.0%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 90.7%

      \[\leadsto \left(\left({x}^{-1} \cdot \left(0.0007936500793651 + y\right)\right) \cdot z\right) \cdot \color{blue}{z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\left(z \cdot z\right) \cdot \left(\frac{1}{x} + \frac{\frac{0.0007936500793651}{x}}{y}\right), y, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;\frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(\left(\log x \cdot x - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + 0.0007936500793651\right) \cdot {x}^{-1}\right) \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.9e+56)
   (+
    (/
     (fma
      (fma (+ y 0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)
     x)
    (fma (- x 0.5) (log x) (- 0.91893853320467 x)))
   (-
    (fma
     (fma (/ y x) z (/ (fma 0.0007936500793651 z -0.0027777777777778) x))
     z
     (fma (- x 0.5) (log x) (/ 0.083333333333333 x)))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.9e+56) {
		tmp = (fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) + fma((x - 0.5), log(x), (0.91893853320467 - x));
	} else {
		tmp = fma(fma((y / x), z, (fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma((x - 0.5), log(x), (0.083333333333333 / x))) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.9e+56)
		tmp = Float64(Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) + fma(Float64(x - 0.5), log(x), Float64(0.91893853320467 - x)));
	else
		tmp = Float64(fma(fma(Float64(y / x), z, Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x)), z, fma(Float64(x - 0.5), log(x), Float64(0.083333333333333 / x))) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.9e+56], N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] * z + N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * z + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.89999999999999994e56

   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. 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%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow-1 N/A

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

      15. associate-/r/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   if 3.89999999999999994e56 < x

   1. Initial program 87.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{y}{x}, z, \frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}\right), z, \mathsf{fma}\left(x - 0.5, \log x, \frac{0.083333333333333}{x}\right)\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 2e+288)
   (+
    (/
     (fma
      (fma (+ y 0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)
     x)
    (fma (- x 0.5) (log x) (- 0.91893853320467 x)))
   (-
    (fma (/ (fma 0.0007936500793651 z -0.0027777777777778) x) z (* (log x) x))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e+288) {
		tmp = (fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) + fma((x - 0.5), log(x), (0.91893853320467 - x));
	} else {
		tmp = fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, (log(x) * x)) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 2e+288)
		tmp = Float64(Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x) + fma(Float64(x - 0.5), log(x), Float64(0.91893853320467 - x)));
	else
		tmp = Float64(fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, Float64(log(x) * x)) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 2e+288], N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $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) < 2e288

   1. Initial program 97.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Add Preprocessing
   3. 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 97.6

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

   4. Applied rewrites 97.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow-1 N/A

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

      15. associate-/r/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

   6. Applied rewrites 97.7%

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

   if 2e288 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 84.4%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right)}{x}, z, -1 \cdot \left(x \cdot \log \left(\frac{1}{x}\right)\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 2 \cdot 10^{+288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 5e+107)
   (+
    (/ (fma (* y z) z 0.083333333333333) x)
    (fma (- x 0.5) (log x) (- 0.91893853320467 x)))
   (-
    (fma (/ (fma 0.0007936500793651 z -0.0027777777777778) x) z (* (log x) x))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 5e+107) {
		tmp = (fma((y * z), z, 0.083333333333333) / x) + fma((x - 0.5), log(x), (0.91893853320467 - x));
	} else {
		tmp = fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, (log(x) * x)) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 5e+107)
		tmp = Float64(Float64(fma(Float64(y * z), z, 0.083333333333333) / x) + fma(Float64(x - 0.5), log(x), Float64(0.91893853320467 - x)));
	else
		tmp = Float64(fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, Float64(log(x) * x)) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 5e+107], N[(N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $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) < 5.0000000000000002e107

   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. 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 97.2

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

   4. Applied rewrites 97.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. sub-neg N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow-1 N/A

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

      15. associate-/r/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

   6. Applied rewrites 97.3%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 96.3

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

   9. Applied rewrites 96.3%

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

   if 5.0000000000000002e107 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.5%

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

   3. Taylor expanded in y around 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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right)}{x}, z, -1 \cdot \left(x \cdot \log \left(\frac{1}{x}\right)\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.7%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} + \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1500000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right) \cdot {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1500000.0)
   (*
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    (pow x -1.0))
   (-
    (fma (/ (fma 0.0007936500793651 z -0.0027777777777778) x) z (* (log x) x))
    (- x 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1500000.0) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) * pow(x, -1.0);
	} else {
		tmp = fma((fma(0.0007936500793651, z, -0.0027777777777778) / x), z, (log(x) * x)) - (x - 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1500000.0)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) * (x ^ -1.0));
	else
		tmp = Float64(fma(Float64(fma(0.0007936500793651, z, -0.0027777777777778) / x), z, Float64(log(x) * x)) - Float64(x - 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1500000.0], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e6

   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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.6%

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

   if 1.5e6 < x

   1. Initial program 89.3%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, z, \frac{-13888888888889}{5000000000000000}\right)}{x}, z, -1 \cdot \left(x \cdot \log \left(\frac{1}{x}\right)\right)\right) - \left(x - \frac{91893853320467}{100000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 83.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1500000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right) \cdot {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right)}{x}, z, \log x \cdot x\right) - \left(x - 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right) \cdot {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3e+53)
   (*
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    (pow x -1.0))
   (* (- (log x) 1.0) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e+53) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) * pow(x, -1.0);
	} else {
		tmp = (log(x) - 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3e+53)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) * (x ^ -1.0));
	else
		tmp = Float64(Float64(log(x) - 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3e+53], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] * N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999998e53

   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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.5%

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

   if 2.99999999999999998e53 < x

   1. Initial program 87.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   4. 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. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 73.3

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

   5. Applied rewrites 73.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right) \cdot {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3e+53)
   (/
    (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 <= 3e+53) {
		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 <= 3e+53)
		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, 3e+53], 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 < 2.99999999999999998e53

   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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 2.99999999999999998e53 < x

   1. Initial program 87.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)} \]
   4. 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. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 73.3

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

   5. Applied rewrites 73.3%

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

Alternative 10: 57.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z\\ t_1 := \left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z))
        (t_1 (* (* (/ z x) z) y)))
   (if (<= t_0 -500000.0)
     t_1
     (if (<= t_0 50.0)
       (/ 0.083333333333333 x)
       (if (<= t_0 5e+132) t_1 (* (* (/ 0.0007936500793651 x) z) z))))))

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) * y;
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = t_1;
	} else if (t_0 <= 50.0) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 5e+132) {
		tmp = t_1;
	} else {
		tmp = ((0.0007936500793651 / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z
    t_1 = ((z / x) * z) * y
    if (t_0 <= (-500000.0d0)) then
        tmp = t_1
    else if (t_0 <= 50.0d0) then
        tmp = 0.083333333333333d0 / x
    else if (t_0 <= 5d+132) then
        tmp = t_1
    else
        tmp = ((0.0007936500793651d0 / x) * z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	double t_1 = ((z / x) * z) * y;
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = t_1;
	} else if (t_0 <= 50.0) {
		tmp = 0.083333333333333 / x;
	} else if (t_0 <= 5e+132) {
		tmp = t_1;
	} else {
		tmp = ((0.0007936500793651 / x) * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z
	t_1 = ((z / x) * z) * y
	tmp = 0
	if t_0 <= -500000.0:
		tmp = t_1
	elif t_0 <= 50.0:
		tmp = 0.083333333333333 / x
	elif t_0 <= 5e+132:
		tmp = t_1
	else:
		tmp = ((0.0007936500793651 / x) * z) * z
	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(Float64(z / x) * z) * y)
	tmp = 0.0
	if (t_0 <= -500000.0)
		tmp = t_1;
	elseif (t_0 <= 50.0)
		tmp = Float64(0.083333333333333 / x);
	elseif (t_0 <= 5e+132)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(0.0007936500793651 / x) * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y + 0.0007936500793651) * z) - 0.0027777777777778) * z;
	t_1 = ((z / x) * z) * y;
	tmp = 0.0;
	if (t_0 <= -500000.0)
		tmp = t_1;
	elseif (t_0 <= 50.0)
		tmp = 0.083333333333333 / x;
	elseif (t_0 <= 5e+132)
		tmp = t_1;
	else
		tmp = ((0.0007936500793651 / x) * z) * z;
	end
	tmp_2 = 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[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -500000.0], t$95$1, If[LessEqual[t$95$0, 50.0], N[(0.083333333333333 / x), $MachinePrecision], If[LessEqual[t$95$0, 5e+132], t$95$1, N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] * z), $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) < -5e5 or 50 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 5.0000000000000001e132

   1. Initial program 94.0%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 67.9

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

   5. Applied rewrites 67.9%

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

   7. Applied rewrites 73.5%

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

   if -5e5 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) < 50

   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 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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 45.8

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.7%

      \[\leadsto \frac{0.083333333333333}{x} \]

   if 5.0000000000000001e132 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 92.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 78.4%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq -500000:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 50:\\ \;\;\;\;\frac{0.083333333333333}{x}\\ \mathbf{elif}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 0.12:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.12)
   (/ (fma (* y z) z 0.083333333333333) x)
   (* (* (/ z x) z) (+ y 0.0007936500793651))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.12) {
		tmp = fma((y * z), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * z) * (y + 0.0007936500793651);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.12)
		tmp = Float64(fma(Float64(y * z), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 0.12], N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $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) < 0.12

   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

   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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot z, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.6%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} \]

   if 0.12 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 24.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{\frac{-1}{4}}\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \frac{-1}{4}\right) \cdot \log x}{\color{blue}{\frac{1}{2} + x}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      12. lower-+.f64 13.3

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{\color{blue}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

   7. Applied rewrites 13.3%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

   8. Taylor expanded in z around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} \cdot {z}^{2} + \frac{y \cdot {z}^{2}}{x} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x}} + \frac{y \cdot {z}^{2}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} + \frac{y \cdot {z}^{2}}{x} \]

      7. associate-/l* N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x} + \color{blue}{y \cdot \frac{{z}^{2}}{x}} \]

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 78.8

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

   10. Applied rewrites 78.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 0.12:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 1e+95)
   (/ (fma (* y z) z 0.083333333333333) x)
   (* (/ (+ y 0.0007936500793651) x) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 1e+95) {
		tmp = fma((y * z), z, 0.083333333333333) / x;
	} else {
		tmp = ((y + 0.0007936500793651) / x) * (z * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 1e+95)
		tmp = Float64(fma(Float64(y * z), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(y + 0.0007936500793651) / x) * Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 1e+95], N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * N[(z * z), $MachinePrecision]), $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) < 1.00000000000000002e95

   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

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

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot z, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.5%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} \]

   if 1.00000000000000002e95 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 48.3

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

   5. Applied rewrites 48.3%

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

   7. Applied rewrites 53.6%

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

   8. Taylor expanded in z around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} \cdot {z}^{2} + \frac{y}{x} \cdot {z}^{2} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x}} + \frac{y}{x} \cdot {z}^{2} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} + \frac{y}{x} \cdot {z}^{2} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x} + \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x} + \color{blue}{y \cdot \frac{{z}^{2}}{x}} \]

      8. distribute-rgt-out N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

          \[\leadsto \color{blue}{{z}^{2} \cdot \frac{\frac{7936500793651}{10000000000000000} + y}{x}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x} \cdot {z}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x} \cdot {z}^{2}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} + y}{x}} \cdot {z}^{2} \]

      14. +-commutative N/A

          \[\leadsto \frac{\color{blue}{y + \frac{7936500793651}{10000000000000000}}}{x} \cdot {z}^{2} \]

      15. lower-+.f64 N/A

          \[\leadsto \frac{\color{blue}{y + \frac{7936500793651}{10000000000000000}}}{x} \cdot {z}^{2} \]

      16. unpow2 N/A

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

      17. lower-*.f64 80.0

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

   10. Applied rewrites 80.0%

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

Alternative 13: 57.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 5e+107)
   (/ (fma (* y z) z 0.083333333333333) x)
   (* (* (/ 0.0007936500793651 x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 5e+107) {
		tmp = fma((y * z), z, 0.083333333333333) / x;
	} else {
		tmp = ((0.0007936500793651 / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 5e+107)
		tmp = Float64(fma(Float64(y * z), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(0.0007936500793651 / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 5e+107], N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] * z), $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) < 5.0000000000000002e107

   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

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

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot z, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.8%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} \]

   if 5.0000000000000002e107 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 89.5%

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

   3. Taylor expanded in y around 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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.6%

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

Alternative 14: 48.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z \leq 0.12:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.12)
   (/ (fma -0.0027777777777778 z 0.083333333333333) x)
   (* (* (/ 0.0007936500793651 x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.12) {
		tmp = fma(-0.0027777777777778, z, 0.083333333333333) / x;
	} else {
		tmp = ((0.0007936500793651 / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) <= 0.12)
		tmp = Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(0.0007936500793651 / x) * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 0.12], N[(N[(-0.0027777777777778 * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] * z), $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) < 0.12

   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

   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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.4%

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

   if 0.12 < (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z)

   1. Initial program 90.9%

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

   3. Taylor expanded in y around 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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.0%

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

Alternative 15: 61.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{if}\;y + 0.0007936500793651 \leq -20000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y + 0.0007936500793651 \leq 0.0007936501:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (fma (* y z) z 0.083333333333333) x)))
   (if (<= (+ y 0.0007936500793651) -20000000.0)
     t_0
     (if (<= (+ y 0.0007936500793651) 0.0007936501)
       (/
        (fma
         (fma 0.0007936500793651 z -0.0027777777777778)
         z
         0.083333333333333)
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y * z), z, 0.083333333333333) / x;
	double tmp;
	if ((y + 0.0007936500793651) <= -20000000.0) {
		tmp = t_0;
	} else if ((y + 0.0007936500793651) <= 0.0007936501) {
		tmp = fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(Float64(y * z), z, 0.083333333333333) / x)
	tmp = 0.0
	if (Float64(y + 0.0007936500793651) <= -20000000.0)
		tmp = t_0;
	elseif (Float64(y + 0.0007936500793651) <= 0.0007936501)
		tmp = Float64(fma(fma(0.0007936500793651, z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], -20000000.0], t$95$0, If[LessEqual[N[(y + 0.0007936500793651), $MachinePrecision], 0.0007936501], N[(N[(N[(0.0007936500793651 * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < -2e7 or 7.93650100000000054e-4 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64))

   1. Initial program 96.0%

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

   3. Taylor expanded in 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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot z, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.9%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} \]

   if -2e7 < (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) < 7.93650100000000054e-4

   1. Initial program 93.0%

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

   3. Taylor expanded in y around 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. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.7%

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

Alternative 16: 64.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.6e+89)
   (/
    (fma
     (fma (+ y 0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ z x) z) (+ y 0.0007936500793651))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.6e+89) {
		tmp = fma(fma((y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * z) * (y + 0.0007936500793651);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.6e+89)
		tmp = Float64(fma(fma(Float64(y + 0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.6e+89], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e89

   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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 2.6000000000000001e89 < x

   1. Initial program 85.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{\frac{-1}{4}}\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \frac{-1}{4}\right) \cdot \log x}{\color{blue}{\frac{1}{2} + x}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      12. lower-+.f64 24.0

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{\color{blue}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

   7. Applied rewrites 24.0%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

   8. Taylor expanded in z around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} \cdot {z}^{2} + \frac{y \cdot {z}^{2}}{x} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x}} + \frac{y \cdot {z}^{2}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} + \frac{y \cdot {z}^{2}}{x} \]

      7. associate-/l* N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x} + \color{blue}{y \cdot \frac{{z}^{2}}{x}} \]

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 27.6

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

   10. Applied rewrites 27.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y + 0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 2.6e+89)
   (/ (fma (* (+ y 0.0007936500793651) z) z 0.083333333333333) x)
   (* (* (/ z x) z) (+ y 0.0007936500793651))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.6e+89) {
		tmp = fma(((y + 0.0007936500793651) * z), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * z) * (y + 0.0007936500793651);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 2.6e+89)
		tmp = Float64(fma(Float64(Float64(y + 0.0007936500793651) * z), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(y + 0.0007936500793651));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 2.6e+89], N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(y + 0.0007936500793651), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e89

   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. metadata-eval N/A

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

      7. *-commutative N/A

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

      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. +-commutative N/A

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

      10. lower-+.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{\mathsf{fma}\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right), z, \frac{83333333333333}{1000000000000000}\right)}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.4%

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

   if 2.6000000000000001e89 < x

   1. Initial program 85.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 74.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{\frac{-1}{4}}\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \frac{-1}{4}\right) \cdot \log x}{\color{blue}{\frac{1}{2} + x}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

      12. lower-+.f64 24.0

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{\color{blue}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

   7. Applied rewrites 24.0%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333}{x} \]

   8. Taylor expanded in z around inf

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

      1. distribute-rgt-in N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{7936500793651}{10000000000000000}}}{x} \cdot {z}^{2} + \frac{y \cdot {z}^{2}}{x} \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{7936500793651}{10000000000000000} \cdot {z}^{2}}{x}} + \frac{y \cdot {z}^{2}}{x} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x}} + \frac{y \cdot {z}^{2}}{x} \]

      7. associate-/l* N/A

         \[\leadsto \frac{7936500793651}{10000000000000000} \cdot \frac{{z}^{2}}{x} + \color{blue}{y \cdot \frac{{z}^{2}}{x}} \]

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 27.6

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

   10. Applied rewrites 27.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y + 0.0007936500793651\right) \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(y + 0.0007936500793651\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (fma -0.0027777777777778 z 0.083333333333333) x))

C

double code(double x, double y, double z) {
	return fma(-0.0027777777777778, z, 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(-0.0027777777777778, z, 0.083333333333333) / x)
end

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in 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. metadata-eval N/A

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

   7. *-commutative N/A

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

   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. +-commutative N/A

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

   10. lower-+.f64 64.2

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

5. Applied rewrites 64.2%

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

6. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]
7. Step-by-step derivation

8. Applied rewrites 24.7%

   \[\leadsto \frac{\mathsf{fma}\left(-0.0027777777777778, z, 0.083333333333333\right)}{x} \]
9. Add Preprocessing

Alternative 19: 23.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 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 = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in 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. metadata-eval N/A

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

   7. *-commutative N/A

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

   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. +-commutative N/A

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

   10. lower-+.f64 64.2

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

5. Applied rewrites 64.2%

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

6. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
7. Step-by-step derivation

8. Applied rewrites 20.2%

   \[\leadsto \frac{0.083333333333333}{x} \]
9. Add Preprocessing

Developer Target 1: 98.6% 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 2024243 
(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)))