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

Percentage Accurate: 93.9% → 99.5%

Time: 13.7s

Alternatives: 16

Speedup: 1.0×

• 30.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 650000:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} + \frac{\left(0.91893853320467 + x\right) \cdot \left(\log x \cdot \mathsf{fma}\left(x, x, -0.25\right)\right) - \left(x \cdot x - 0.8444480278083504\right) \cdot \left(0.5 + x\right)}{\left(0.5 + x\right) \cdot \left(0.91893853320467 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 650000.0)
   (+
    (/
     (+
      0.083333333333333
      (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
     x)
    (/
     (-
      (* (+ 0.91893853320467 x) (* (log x) (fma x x -0.25)))
      (* (- (* x x) 0.8444480278083504) (+ 0.5 x)))
     (* (+ 0.5 x) (+ 0.91893853320467 x))))
   (+
    (* (* (/ z x) z) (+ 0.0007936500793651 y))
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 650000.0) {
		tmp = ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x) + ((((0.91893853320467 + x) * (log(x) * fma(x, x, -0.25))) - (((x * x) - 0.8444480278083504) * (0.5 + x))) / ((0.5 + x) * (0.91893853320467 + x)));
	} else {
		tmp = (((z / x) * z) * (0.0007936500793651 + y)) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 650000.0)
		tmp = Float64(Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x) + Float64(Float64(Float64(Float64(0.91893853320467 + x) * Float64(log(x) * fma(x, x, -0.25))) - Float64(Float64(Float64(x * x) - 0.8444480278083504) * Float64(0.5 + x))) / Float64(Float64(0.5 + x) * Float64(0.91893853320467 + x))));
	else
		tmp = Float64(Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y)) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 650000.0], N[(N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(N[(0.91893853320467 + x), $MachinePrecision] * N[(N[Log[x], $MachinePrecision] * N[(x * x + -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x * x), $MachinePrecision] - 0.8444480278083504), $MachinePrecision] * N[(0.5 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 + x), $MachinePrecision] * N[(0.91893853320467 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e5

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-*l/ N/A

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

      8. flip-- N/A

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

      9. frac-sub N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x\right) \cdot \left(x + 0.91893853320467\right) - \left(0.5 + x\right) \cdot \left(x \cdot x - 0.8444480278083504\right)}{\left(0.5 + x\right) \cdot \left(x + 0.91893853320467\right)}} + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 6.5e5 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in 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 63.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.5

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

   8. Applied rewrites 88.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 650000:\\ \;\;\;\;\frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} + \frac{\left(0.91893853320467 + x\right) \cdot \left(\log x \cdot \mathsf{fma}\left(x, x, -0.25\right)\right) - \left(x \cdot x - 0.8444480278083504\right) \cdot \left(0.5 + x\right)}{\left(0.5 + x\right) \cdot \left(0.91893853320467 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            0.083333333333333
            (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
           x))))
   (if (<= t_0 -4e+125)
     (* (+ (/ 0.0007936500793651 x) (/ y x)) (* z z))
     (if (<= t_0 5e+301)
       (fma
        (- x 0.5)
        (log x)
        (- (+ (/ 0.083333333333333 x) 0.91893853320467) x))
       (* (* (/ z x) z) (+ 0.0007936500793651 y))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	double tmp;
	if (t_0 <= -4e+125) {
		tmp = ((0.0007936500793651 / x) + (y / x)) * (z * z);
	} else if (t_0 <= 5e+301) {
		tmp = fma((x - 0.5), log(x), (((0.083333333333333 / x) + 0.91893853320467) - x));
	} else {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x))
	tmp = 0.0
	if (t_0 <= -4e+125)
		tmp = Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * Float64(z * z));
	elseif (t_0 <= 5e+301)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(Float64(0.083333333333333 / x) + 0.91893853320467) - x));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+125], N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+301], 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[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $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)) < -3.9999999999999997e125

   1. Initial program 91.2%

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

   3. Taylor expanded in 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 98.3

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

   5. Applied rewrites 98.3%

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

   if -3.9999999999999997e125 < (+.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)) < 5.0000000000000004e301

   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 87.3

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

   5. Applied rewrites 87.3%

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

   if 5.0000000000000004e301 < (+.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 81.2%

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

   3. Taylor expanded in 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 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around inf

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

       1. 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. 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 90.3

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

   11. Applied rewrites 90.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(\frac{0.083333333333333}{x} + 0.91893853320467\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            0.083333333333333
            (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
           x))))
   (if (<= t_0 -4e+125)
     (* (+ (/ 0.0007936500793651 x) (/ y x)) (* z z))
     (if (<= t_0 5e+301)
       (+
        (fma (- x 0.5) (log x) 0.91893853320467)
        (- (/ 0.083333333333333 x) x))
       (* (* (/ z x) z) (+ 0.0007936500793651 y))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	double tmp;
	if (t_0 <= -4e+125) {
		tmp = ((0.0007936500793651 / x) + (y / x)) * (z * z);
	} else if (t_0 <= 5e+301) {
		tmp = fma((x - 0.5), log(x), 0.91893853320467) + ((0.083333333333333 / x) - x);
	} else {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x))
	tmp = 0.0
	if (t_0 <= -4e+125)
		tmp = Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * Float64(z * z));
	elseif (t_0 <= 5e+301)
		tmp = Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) + Float64(Float64(0.083333333333333 / x) - x));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+125], N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+301], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $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)) < -3.9999999999999997e125

   1. Initial program 91.2%

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

   3. Taylor expanded in 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 98.3

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

   5. Applied rewrites 98.3%

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

   if -3.9999999999999997e125 < (+.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)) < 5.0000000000000004e301

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot {z}^{2}}{x}} \]
   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 8.9

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

   5. Applied rewrites 8.9%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. associate-+r+ N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower--.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-log.f64 87.2

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

   8. Applied rewrites 87.2%

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

   if 5.0000000000000004e301 < (+.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 81.2%

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

   3. Taylor expanded in 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 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around inf

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

       1. 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. 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 90.3

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

   11. Applied rewrites 90.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) + \left(\frac{0.083333333333333}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(\left(\log x \cdot x - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            0.083333333333333
            (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
           x))))
   (if (<= t_0 -4e+125)
     (* (+ (/ 0.0007936500793651 x) (/ y x)) (* z z))
     (if (<= t_0 5e+301)
       (+ (/ 0.083333333333333 x) (+ (- (* (log x) x) x) 0.91893853320467))
       (* (* (/ z x) z) (+ 0.0007936500793651 y))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	double tmp;
	if (t_0 <= -4e+125) {
		tmp = ((0.0007936500793651 / x) + (y / x)) * (z * z);
	} else if (t_0 <= 5e+301) {
		tmp = (0.083333333333333 / x) + (((log(x) * x) - x) + 0.91893853320467);
	} else {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((0.083333333333333d0 + (((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0) * z)) / x)
    if (t_0 <= (-4d+125)) then
        tmp = ((0.0007936500793651d0 / x) + (y / x)) * (z * z)
    else if (t_0 <= 5d+301) then
        tmp = (0.083333333333333d0 / x) + (((log(x) * x) - x) + 0.91893853320467d0)
    else
        tmp = ((z / x) * z) * (0.0007936500793651d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	double tmp;
	if (t_0 <= -4e+125) {
		tmp = ((0.0007936500793651 / x) + (y / x)) * (z * z);
	} else if (t_0 <= 5e+301) {
		tmp = (0.083333333333333 / x) + (((Math.log(x) * x) - x) + 0.91893853320467);
	} else {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x)
	tmp = 0
	if t_0 <= -4e+125:
		tmp = ((0.0007936500793651 / x) + (y / x)) * (z * z)
	elif t_0 <= 5e+301:
		tmp = (0.083333333333333 / x) + (((math.log(x) * x) - x) + 0.91893853320467)
	else:
		tmp = ((z / x) * z) * (0.0007936500793651 + y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x))
	tmp = 0.0
	if (t_0 <= -4e+125)
		tmp = Float64(Float64(Float64(0.0007936500793651 / x) + Float64(y / x)) * Float64(z * z));
	elseif (t_0 <= 5e+301)
		tmp = Float64(Float64(0.083333333333333 / x) + Float64(Float64(Float64(log(x) * x) - x) + 0.91893853320467));
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	tmp = 0.0;
	if (t_0 <= -4e+125)
		tmp = ((0.0007936500793651 / x) + (y / x)) * (z * z);
	elseif (t_0 <= 5e+301)
		tmp = (0.083333333333333 / x) + (((log(x) * x) - x) + 0.91893853320467);
	else
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+125], N[(N[(N[(0.0007936500793651 / x), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+301], N[(N[(0.083333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $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)) < -3.9999999999999997e125

   1. Initial program 91.2%

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

   3. Taylor expanded in 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 98.3

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

   5. Applied rewrites 98.3%

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

   if -3.9999999999999997e125 < (+.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)) < 5.0000000000000004e301

   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 87.2%

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

   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 85.1

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

   8. Applied rewrites 85.1%

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

   if 5.0000000000000004e301 < (+.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 81.2%

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

   3. Taylor expanded in 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 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around inf

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

       1. 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. 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 90.3

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

   11. Applied rewrites 90.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{0.0007936500793651}{x} + \frac{y}{x}\right) \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x} \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{0.083333333333333}{x} + \left(\left(\log x \cdot x - x\right) + 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 650000:\\ \;\;\;\;\left(\left(\frac{\log x \cdot \mathsf{fma}\left(x, x, -0.25\right)}{0.5 + x} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 650000.0)
   (+
    (+ (- (/ (* (log x) (fma x x -0.25)) (+ 0.5 x)) x) 0.91893853320467)
    (/
     (+
      0.083333333333333
      (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
     x))
   (+
    (* (* (/ z x) z) (+ 0.0007936500793651 y))
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 650000.0) {
		tmp = ((((log(x) * fma(x, x, -0.25)) / (0.5 + x)) - x) + 0.91893853320467) + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	} else {
		tmp = (((z / x) * z) * (0.0007936500793651 + y)) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 650000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(log(x) * fma(x, x, -0.25)) / Float64(0.5 + x)) - x) + 0.91893853320467) + Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x));
	else
		tmp = Float64(Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y)) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 650000.0], N[(N[(N[(N[(N[(N[Log[x], $MachinePrecision] * N[(x * x + -0.25), $MachinePrecision]), $MachinePrecision] / N[(0.5 + x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e5

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      10. metadata-eval N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{\frac{-1}{4}}\right) \cdot \log x}{x + \frac{1}{2}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, \frac{-1}{4}\right) \cdot \log x}{\color{blue}{\frac{1}{2} + x}} - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      12. lower-+.f64 99.7

          \[\leadsto \left(\left(\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{\color{blue}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied rewrites 99.7%

      \[\leadsto \left(\left(\color{blue}{\frac{\mathsf{fma}\left(x, x, -0.25\right) \cdot \log x}{0.5 + x}} - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   if 6.5e5 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in 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 63.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.5

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

   8. Applied rewrites 88.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 650000:\\ \;\;\;\;\left(\left(\frac{\log x \cdot \mathsf{fma}\left(x, x, -0.25\right)}{0.5 + x} - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\\ \mathbf{if}\;x \leq 650000:\\ \;\;\;\;t\_0 + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))
   (if (<= x 650000.0)
     (+
      t_0
      (/
       (+
        0.083333333333333
        (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z))
       x))
     (+ (* (* (/ z x) z) (+ 0.0007936500793651 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	double tmp;
	if (x <= 650000.0) {
		tmp = t_0 + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	} else {
		tmp = (((z / x) * z) * (0.0007936500793651 + y)) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0
    if (x <= 650000.0d0) then
        tmp = t_0 + ((0.083333333333333d0 + (((z * (0.0007936500793651d0 + y)) - 0.0027777777777778d0) * z)) / x)
    else
        tmp = (((z / x) * z) * (0.0007936500793651d0 + y)) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x - 0.5) * Math.log(x)) - x) + 0.91893853320467;
	double tmp;
	if (x <= 650000.0) {
		tmp = t_0 + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	} else {
		tmp = (((z / x) * z) * (0.0007936500793651 + y)) + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x - 0.5) * math.log(x)) - x) + 0.91893853320467
	tmp = 0
	if x <= 650000.0:
		tmp = t_0 + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x)
	else:
		tmp = (((z / x) * z) * (0.0007936500793651 + y)) + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467)
	tmp = 0.0
	if (x <= 650000.0)
		tmp = Float64(t_0 + Float64(Float64(0.083333333333333 + Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x));
	else
		tmp = Float64(Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x - 0.5) * log(x)) - x) + 0.91893853320467;
	tmp = 0.0;
	if (x <= 650000.0)
		tmp = t_0 + ((0.083333333333333 + (((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z)) / x);
	else
		tmp = (((z / x) * z) * (0.0007936500793651 + y)) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]}, If[LessEqual[x, 650000.0], N[(t$95$0 + N[(N[(0.083333333333333 + N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e5

   1. Initial program 99.7%

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

   if 6.5e5 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in 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 63.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.5

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

   8. Applied rewrites 88.5%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 650000:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{0.083333333333333 + \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+183}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x} + \left(\log x - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00043)
   (/
    (fma
     (fma -0.5 (log x) 0.91893853320467)
     x
     (fma
      (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
      z
      0.083333333333333))
    x)
   (if (<= x 1.02e+183)
     (+ (* (* z z) (/ (+ 0.0007936500793651 y) x)) (* (- (log x) 1.0) x))
     (+
      (* (* (/ z x) z) y)
      (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00043) {
		tmp = fma(fma(-0.5, log(x), 0.91893853320467), x, fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333)) / x;
	} else if (x <= 1.02e+183) {
		tmp = ((z * z) * ((0.0007936500793651 + y) / x)) + ((log(x) - 1.0) * x);
	} else {
		tmp = (((z / x) * z) * y) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00043)
		tmp = Float64(fma(fma(-0.5, log(x), 0.91893853320467), x, fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333)) / x);
	elseif (x <= 1.02e+183)
		tmp = Float64(Float64(Float64(z * z) * Float64(Float64(0.0007936500793651 + y) / x)) + Float64(Float64(log(x) - 1.0) * x));
	else
		tmp = Float64(Float64(Float64(Float64(z / x) * z) * y) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00043], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.02e+183], N[(N[(N[(z * z), $MachinePrecision] * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.29999999999999989e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 99.3%

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

   if 4.29999999999999989e-4 < x < 1.02000000000000002e183

   1. Initial program 91.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 55.6%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 96.1

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

   8. Applied rewrites 96.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. mul-1-neg N/A

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

       3. log-rec N/A

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

       4. remove-double-neg N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-log.f64 95.4

          \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x + \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right) \]

   11. Applied rewrites 95.4%

       \[\leadsto \color{blue}{\left(\log x - 1\right) \cdot x} + \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right) \]

   if 1.02000000000000002e183 < x

   1. Initial program 69.8%

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

   3. Taylor expanded in 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 76.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{{z}^{2}}{x} \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 89.8%

       \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \left(z \cdot \frac{z}{x}\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+183}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x} + \left(\log x - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot y + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00043)
   (/
    (fma
     (fma -0.5 (log x) 0.91893853320467)
     x
     (fma
      (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
      z
      0.083333333333333))
    x)
   (+
    (* (* (/ z x) z) (+ 0.0007936500793651 y))
    (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00043) {
		tmp = fma(fma(-0.5, log(x), 0.91893853320467), x, fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333)) / x;
	} else {
		tmp = (((z / x) * z) * (0.0007936500793651 + y)) + ((((x - 0.5) * log(x)) - x) + 0.91893853320467);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00043)
		tmp = Float64(fma(fma(-0.5, log(x), 0.91893853320467), x, fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333)) / x);
	else
		tmp = Float64(Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y)) + Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00043], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.29999999999999989e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 99.3%

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

   if 4.29999999999999989e-4 < x

   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 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 62.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.1

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

   8. Applied rewrites 88.1%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \color{blue}{\frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) + \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x} + \left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00043)
   (/
    (fma
     (fma -0.5 (log x) 0.91893853320467)
     x
     (fma
      (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
      z
      0.083333333333333))
    x)
   (+ (* (* z z) (/ (+ 0.0007936500793651 y) x)) (* (- (log x) 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00043) {
		tmp = fma(fma(-0.5, log(x), 0.91893853320467), x, fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333)) / x;
	} else {
		tmp = ((z * z) * ((0.0007936500793651 + y) / x)) + ((log(x) - 1.0) * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00043)
		tmp = Float64(fma(fma(-0.5, log(x), 0.91893853320467), x, fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333)) / x);
	else
		tmp = Float64(Float64(Float64(z * z) * Float64(Float64(0.0007936500793651 + y) / x)) + Float64(Float64(log(x) - 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00043], N[(N[(N[(-0.5 * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] * x + N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.29999999999999989e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 99.3%

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

   if 4.29999999999999989e-4 < x

   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 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 62.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.1

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

   8. Applied rewrites 88.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. mul-1-neg N/A

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

       3. log-rec N/A

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

       4. remove-double-neg N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-log.f64 87.7

          \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x + \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right) \]

   11. Applied rewrites 87.7%

       \[\leadsto \color{blue}{\left(\log x - 1\right) \cdot x} + \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \log x, 0.91893853320467\right), x, \mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x} + \left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x} + \left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00043)
   (+
    (/
     (fma
      (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
      z
      0.083333333333333)
     x)
    (- 0.91893853320467 x))
   (+ (* (* z z) (/ (+ 0.0007936500793651 y) x)) (* (- (log x) 1.0) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00043) {
		tmp = (fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + (0.91893853320467 - x);
	} else {
		tmp = ((z * z) * ((0.0007936500793651 + y) / x)) + ((log(x) - 1.0) * x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00043)
		tmp = Float64(Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x) + Float64(0.91893853320467 - x));
	else
		tmp = Float64(Float64(Float64(z * z) * Float64(Float64(0.0007936500793651 + y) / x)) + Float64(Float64(log(x) - 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00043], N[(N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] + N[(0.91893853320467 - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * z), $MachinePrecision] * N[(N[(0.0007936500793651 + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[x], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.29999999999999989e-4

   1. Initial program 99.7%

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

   3. 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 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.6%

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

   if 4.29999999999999989e-4 < x

   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 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 62.4%

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

   6. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.1

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

   8. Applied rewrites 88.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. mul-1-neg N/A

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

       3. log-rec N/A

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

       4. remove-double-neg N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-log.f64 87.7

          \[\leadsto \left(\color{blue}{\log x} - 1\right) \cdot x + \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right) \]

   11. Applied rewrites 87.7%

       \[\leadsto \color{blue}{\left(\log x - 1\right) \cdot x} + \frac{y + 0.0007936500793651}{x} \cdot \left(z \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00043:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} + \left(0.91893853320467 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.0007936500793651 + y}{x} + \left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.7e+28)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) 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 <= 3.7e+28) {
		tmp = fma(fma((0.0007936500793651 + y), 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 <= 3.7e+28)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), 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, 3.7e+28], N[(N[(N[(N[(0.0007936500793651 + y), $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 < 3.6999999999999999e28

   1. Initial program 99.7%

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

   3. 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 96.2

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

   5. Applied rewrites 96.2%

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

   if 3.6999999999999999e28 < x

   1. Initial program 82.7%

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

   3. Taylor expanded in 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 65.6

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

   5. Applied rewrites 65.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{x} \cdot z\\ \mathbf{if}\;\left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(t\_0, y, \frac{1}{12.000000000000048 \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z x) z)))
   (if (<= (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z) 5.0)
     (fma t_0 y (/ 1.0 (* 12.000000000000048 x)))
     (* t_0 (+ 0.0007936500793651 y)))))

C

double code(double x, double y, double z) {
	double t_0 = (z / x) * z;
	double tmp;
	if ((((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z) <= 5.0) {
		tmp = fma(t_0, y, (1.0 / (12.000000000000048 * x)));
	} else {
		tmp = t_0 * (0.0007936500793651 + y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / x) * z)
	tmp = 0.0
	if (Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z) <= 5.0)
		tmp = fma(t_0, y, Float64(1.0 / Float64(12.000000000000048 * x)));
	else
		tmp = Float64(t_0 * Float64(0.0007936500793651 + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 5.0], N[(t$95$0 * y + N[(1.0 / N[(12.000000000000048 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.0007936500793651 + y), $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

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.3%

      \[\leadsto \mathsf{fma}\left(\frac{z}{x} \cdot z, y, \frac{0.083333333333333}{x}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 65.3%

       \[\leadsto \mathsf{fma}\left(\frac{z}{x} \cdot z, y, \frac{1}{x \cdot 12.000000000000048}\right) \]

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

   1. Initial program 86.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in z around inf

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

       1. 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. 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 76.7

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

   11. Applied rewrites 76.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right) \cdot z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{x} \cdot z, y, \frac{1}{12.000000000000048 \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.9% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z x) z)))
   (if (<= (* (- (* z (+ 0.0007936500793651 y)) 0.0027777777777778) z) 5.0)
     (fma t_0 y (/ 0.083333333333333 x))
     (* t_0 (+ 0.0007936500793651 y)))))

C

double code(double x, double y, double z) {
	double t_0 = (z / x) * z;
	double tmp;
	if ((((z * (0.0007936500793651 + y)) - 0.0027777777777778) * z) <= 5.0) {
		tmp = fma(t_0, y, (0.083333333333333 / x));
	} else {
		tmp = t_0 * (0.0007936500793651 + y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / x) * z)
	tmp = 0.0
	if (Float64(Float64(Float64(z * Float64(0.0007936500793651 + y)) - 0.0027777777777778) * z) <= 5.0)
		tmp = fma(t_0, y, Float64(0.083333333333333 / x));
	else
		tmp = Float64(t_0 * Float64(0.0007936500793651 + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision], 5.0], N[(t$95$0 * y + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(0.0007936500793651 + y), $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

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.3%

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

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

   1. Initial program 86.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in z around inf

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

       1. 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. 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 76.7

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

   11. Applied rewrites 76.7%

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

4. Final simplification 70.7%

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

Alternative 14: 65.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.25e+60)
   (/
    (fma
     (fma (+ 0.0007936500793651 y) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ z x) z) (+ 0.0007936500793651 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.25e+60) {
		tmp = fma(fma((0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = ((z / x) * z) * (0.0007936500793651 + y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.25e+60)
		tmp = Float64(fma(fma(Float64(0.0007936500793651 + y), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.25e+60], N[(N[(N[(N[(0.0007936500793651 + y), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(z / x), $MachinePrecision] * z), $MachinePrecision] * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.24999999999999994e60

   1. Initial program 99.7%

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

   3. 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 92.1

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

   5. Applied rewrites 92.1%

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

   if 1.24999999999999994e60 < x

   1. Initial program 81.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   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 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in z around inf

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

       1. 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. 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 38.2

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

   11. Applied rewrites 38.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0007936500793651 + y, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* (/ z x) z) (+ 0.0007936500793651 y)))

C

double code(double x, double y, double z) {
	return ((z / x) * z) * (0.0007936500793651 + y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((z / x) * z) * (0.0007936500793651d0 + y)
end function

Java

public static double code(double x, double y, double z) {
	return ((z / x) * z) * (0.0007936500793651 + y);
}

Python

def code(x, y, z):
	return ((z / x) * z) * (0.0007936500793651 + y)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(z / x) * z) * Float64(0.0007936500793651 + y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((z / x) * z) * (0.0007936500793651 + y);
end

Wolfram

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

Derivation

1. Initial program 92.5%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 34.7

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

5. Applied rewrites 34.7%

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

6. Taylor expanded in z around 0

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

   1. sub-neg N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f64 N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

8. Applied rewrites 93.3%

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

9. Taylor expanded in z around inf

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

    1. 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. 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 49.4

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

11. Applied rewrites 49.4%

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

12. Final simplification 49.4%

    \[\leadsto \left(\frac{z}{x} \cdot z\right) \cdot \left(0.0007936500793651 + y\right) \]
13. Add Preprocessing

Alternative 16: 31.8% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 34.7

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

5. Applied rewrites 34.7%

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

7. Applied rewrites 38.5%

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

8. Final simplification 38.5%

   \[\leadsto \left(\frac{z}{x} \cdot z\right) \cdot y \]
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 2024268 
(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)))