Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B

Percentage Accurate: 99.9% → 99.9%

Time: 10.2s

Alternatives: 13

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}{1.4142071247754946} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- (/ (fma 0.27061 x 2.30753) (fma (fma 0.04481 x 0.99229) x 1.0)) x)
  1.4142071247754946))

C

double code(double x) {
	return ((fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x) / 1.4142071247754946;
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x) / 1.4142071247754946)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.27061 * x + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / 1.4142071247754946), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   3. flip-- N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}} \]

   4. clear-num N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{1}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.70711}{\frac{1}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}{\frac{70711}{100000}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}{\frac{70711}{100000}}}} \]

   4. div-inv N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x} \cdot \frac{1}{\frac{70711}{100000}}}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x} \cdot \frac{1}{\frac{70711}{100000}}}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}} \cdot \frac{1}{\frac{70711}{100000}}} \]

   7. inv-pow N/A

      \[\leadsto \frac{1}{\color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1}} \cdot \frac{1}{\frac{70711}{100000}}} \]

   8. lower-pow.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{\left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1}} \cdot \frac{1}{\frac{70711}{100000}}} \]

   9. lift-fma.f64 N/A

      \[\leadsto \frac{1}{{\left(\frac{\color{blue}{\frac{27061}{100000} \cdot x + \frac{230753}{100000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1} \cdot \frac{1}{\frac{70711}{100000}}} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{{\left(\frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1} \cdot \frac{1}{\frac{70711}{100000}}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1}{{\left(\frac{\color{blue}{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1} \cdot \frac{1}{\frac{70711}{100000}}} \]

   12. metadata-eval 99.7

       \[\leadsto \frac{1}{{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right)}^{-1} \cdot \color{blue}{1.4142071247754946}} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right)}^{-1} \cdot 1.4142071247754946}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{{\left(\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1} \cdot \frac{100000}{70711}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{\left(\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1} \cdot \frac{100000}{70711}}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{{\left(\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1}}}{\frac{100000}{70711}}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{{\left(\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)}^{-1}}}}{\frac{100000}{70711}} \]

   5. unpow-1 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}}}{\frac{100000}{70711}} \]

   6. remove-double-div N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}{\frac{100000}{70711}} \]

   7. lower-/.f64 99.9

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}{1.4142071247754946}} \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}{\frac{100000}{70711}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}{\frac{100000}{70711}} \]

   10. lower-fma.f64 99.9

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}{1.4142071247754946} \]

8. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}{1.4142071247754946}} \]
9. Add Preprocessing

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\frac{0.70711}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.373935803238505, x, 0.38500059084694704\right), x, 0.5670050797822634\right), x, 0.4333638132548656\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{4.2702753202410175}{x \cdot x} - 0.70711\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* x 0.27061) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 2.0)
     (fma -0.70711 x (/ 4.2702753202410175 x))
     (if (<= t_0 4.0)
       (/
        0.70711
        (fma
         (fma
          (fma 0.373935803238505 x 0.38500059084694704)
          x
          0.5670050797822634)
         x
         0.4333638132548656))
       (* (- (/ 4.2702753202410175 (* x x)) 0.70711) x)))))

C

double code(double x) {
	double t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = fma(-0.70711, x, (4.2702753202410175 / x));
	} else if (t_0 <= 4.0) {
		tmp = 0.70711 / fma(fma(fma(0.373935803238505, x, 0.38500059084694704), x, 0.5670050797822634), x, 0.4333638132548656);
	} else {
		tmp = ((4.2702753202410175 / (x * x)) - 0.70711) * x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = fma(-0.70711, x, Float64(4.2702753202410175 / x));
	elseif (t_0 <= 4.0)
		tmp = Float64(0.70711 / fma(fma(fma(0.373935803238505, x, 0.38500059084694704), x, 0.5670050797822634), x, 0.4333638132548656));
	else
		tmp = Float64(Float64(Float64(4.2702753202410175 / Float64(x * x)) - 0.70711) * x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(0.70711 / N[(N[(N[(0.373935803238505 * x + 0.38500059084694704), $MachinePrecision] * x + 0.5670050797822634), $MachinePrecision] * x + 0.4333638132548656), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.2702753202410175 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.70711), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. remove-double-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-70711}{100000}, x, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \]

   if 2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. flip-- N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}} \]

      4. clear-num N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{1}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{0.70711}{\frac{1}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{70711}{100000}}{\color{blue}{\frac{100000}{230753} + x \cdot \left(\frac{30191289437}{53246947009} + x \cdot \left(\frac{4730460973492672}{12286892763167777} + \frac{1060196761744488379977}{2835237365779254046081} \cdot x\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\color{blue}{x \cdot \left(\frac{30191289437}{53246947009} + x \cdot \left(\frac{4730460973492672}{12286892763167777} + \frac{1060196761744488379977}{2835237365779254046081} \cdot x\right)\right) + \frac{100000}{230753}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\color{blue}{\left(\frac{30191289437}{53246947009} + x \cdot \left(\frac{4730460973492672}{12286892763167777} + \frac{1060196761744488379977}{2835237365779254046081} \cdot x\right)\right) \cdot x} + \frac{100000}{230753}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\color{blue}{\mathsf{fma}\left(\frac{30191289437}{53246947009} + x \cdot \left(\frac{4730460973492672}{12286892763167777} + \frac{1060196761744488379977}{2835237365779254046081} \cdot x\right), x, \frac{100000}{230753}\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{4730460973492672}{12286892763167777} + \frac{1060196761744488379977}{2835237365779254046081} \cdot x\right) + \frac{30191289437}{53246947009}}, x, \frac{100000}{230753}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\mathsf{fma}\left(\color{blue}{\left(\frac{4730460973492672}{12286892763167777} + \frac{1060196761744488379977}{2835237365779254046081} \cdot x\right) \cdot x} + \frac{30191289437}{53246947009}, x, \frac{100000}{230753}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{4730460973492672}{12286892763167777} + \frac{1060196761744488379977}{2835237365779254046081} \cdot x, x, \frac{30191289437}{53246947009}\right)}, x, \frac{100000}{230753}\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1060196761744488379977}{2835237365779254046081} \cdot x + \frac{4730460973492672}{12286892763167777}}, x, \frac{30191289437}{53246947009}\right), x, \frac{100000}{230753}\right)} \]

      8. lower-fma.f64 99.6

         \[\leadsto \frac{0.70711}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.373935803238505, x, 0.38500059084694704\right)}, x, 0.5670050797822634\right), x, 0.4333638132548656\right)} \]

   7. Applied rewrites 99.6%

      \[\leadsto \frac{0.70711}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.373935803238505, x, 0.38500059084694704\right), x, 0.5670050797822634\right), x, 0.4333638132548656\right)}} \]

   if 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.6%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. flip-- N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}} \]

      4. clear-num N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{1}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{0.70711}{\frac{1}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{1} \cdot \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{70711}{100000}} \cdot \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000}} \]

      6. lower-*.f64 99.6

         \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711} \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{27061}{100000} \cdot x + \frac{230753}{100000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

      9. lower-fma.f64 99.6

         \[\leadsto \left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711 \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right) \cdot x} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \cdot x \]

      3. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right) \cdot x \]

      4. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) + \frac{70711}{100000}\right)\right)\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right)\right) \cdot x \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right)\right) \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot x} \]

      8. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right)\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) + \frac{70711}{100000}\right)}\right)\right) \cdot x \]

      10. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \cdot x \]

      11. remove-double-neg N/A

          \[\leadsto \left(\color{blue}{\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right) \cdot x \]

      12. sub-neg N/A

          \[\leadsto \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \cdot x \]

      13. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \cdot x \]

      14. associate-*r/ N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{1913510371}{448100000} \cdot 1}{{x}^{2}}} - \frac{70711}{100000}\right) \cdot x \]

      15. metadata-eval N/A

          \[\leadsto \left(\frac{\color{blue}{\frac{1913510371}{448100000}}}{{x}^{2}} - \frac{70711}{100000}\right) \cdot x \]

      16. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{1913510371}{448100000}}{{x}^{2}}} - \frac{70711}{100000}\right) \cdot x \]

      17. unpow2 N/A

          \[\leadsto \left(\frac{\frac{1913510371}{448100000}}{\color{blue}{x \cdot x}} - \frac{70711}{100000}\right) \cdot x \]

      18. lower-*.f64 98.1

          \[\leadsto \left(\frac{4.2702753202410175}{\color{blue}{x \cdot x}} - 0.70711\right) \cdot x \]

   9. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(\frac{4.2702753202410175}{x \cdot x} - 0.70711\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;\frac{0.70711}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.373935803238505, x, 0.38500059084694704\right), x, 0.5670050797822634\right), x, 0.4333638132548656\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{4.2702753202410175}{x \cdot x} - 0.70711\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{4.2702753202410175}{x \cdot x} - 0.70711\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* x 0.27061) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 2.0)
     (fma -0.70711 x (/ 4.2702753202410175 x))
     (if (<= t_0 4.0)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       (* (- (/ 4.2702753202410175 (* x x)) 0.70711) x)))))

C

double code(double x) {
	double t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = fma(-0.70711, x, (4.2702753202410175 / x));
	} else if (t_0 <= 4.0) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = ((4.2702753202410175 / (x * x)) - 0.70711) * x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = fma(-0.70711, x, Float64(4.2702753202410175 / x));
	elseif (t_0 <= 4.0)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = Float64(Float64(Float64(4.2702753202410175 / Float64(x * x)) - 0.70711) * x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(N[(N[(4.2702753202410175 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.70711), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. remove-double-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-70711}{100000}, x, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \]

   if 2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \cdot x} + \frac{16316775383}{10000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, x, \frac{16316775383}{10000000000}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, x, \frac{16316775383}{10000000000}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, x, \frac{-2134856267379707}{1000000000000000}\right)}, x, \frac{16316775383}{10000000000}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}}, x, \frac{-2134856267379707}{1000000000000000}\right), x, \frac{16316775383}{10000000000}\right) \]

      9. lower-fma.f64 99.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right)}, x, -2.134856267379707\right), x, 1.6316775383\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)} \]

   if 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.6%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      3. flip-- N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}} \]

      4. clear-num N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{1}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{0.70711}{\frac{1}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{70711}{100000}}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{1} \cdot \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{70711}{100000}} \cdot \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000}} \]

      6. lower-*.f64 99.6

         \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711} \]

      7. lift-fma.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{27061}{100000} \cdot x + \frac{230753}{100000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

      8. *-commutative N/A

         \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

      9. lower-fma.f64 99.6

         \[\leadsto \left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711 \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right) \cdot x} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \cdot x \]

      3. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right) \cdot x \]

      4. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) + \frac{70711}{100000}\right)\right)\right)} \cdot x \]

      5. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right)\right) \cdot x \]

      6. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right)\right) \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{70711}{100000} - \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot x} \]

      8. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{70711}{100000} + \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right)\right) \cdot x \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) + \frac{70711}{100000}\right)}\right)\right) \cdot x \]

      10. distribute-neg-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \cdot x \]

      11. remove-double-neg N/A

          \[\leadsto \left(\color{blue}{\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right) \cdot x \]

      12. sub-neg N/A

          \[\leadsto \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \cdot x \]

      13. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \cdot x \]

      14. associate-*r/ N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{1913510371}{448100000} \cdot 1}{{x}^{2}}} - \frac{70711}{100000}\right) \cdot x \]

      15. metadata-eval N/A

          \[\leadsto \left(\frac{\color{blue}{\frac{1913510371}{448100000}}}{{x}^{2}} - \frac{70711}{100000}\right) \cdot x \]

      16. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{1913510371}{448100000}}{{x}^{2}}} - \frac{70711}{100000}\right) \cdot x \]

      17. unpow2 N/A

          \[\leadsto \left(\frac{\frac{1913510371}{448100000}}{\color{blue}{x \cdot x}} - \frac{70711}{100000}\right) \cdot x \]

      18. lower-*.f64 98.1

          \[\leadsto \left(\frac{4.2702753202410175}{\color{blue}{x \cdot x}} - 0.70711\right) \cdot x \]

   9. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(\frac{4.2702753202410175}{x \cdot x} - 0.70711\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{4.2702753202410175}{x \cdot x} - 0.70711\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - x\right) \cdot 0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* x 0.27061) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 2.0)
     (fma -0.70711 x (/ 4.2702753202410175 x))
     (if (<= t_0 4.0)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       (* (- (/ 6.039053782637804 x) x) 0.70711)))))

C

double code(double x) {
	double t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = fma(-0.70711, x, (4.2702753202410175 / x));
	} else if (t_0 <= 4.0) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = ((6.039053782637804 / x) - x) * 0.70711;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = fma(-0.70711, x, Float64(4.2702753202410175 / x));
	elseif (t_0 <= 4.0)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = Float64(Float64(Float64(6.039053782637804 / x) - x) * 0.70711);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. remove-double-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-70711}{100000}, x, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \]

   if 2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \cdot x} + \frac{16316775383}{10000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, x, \frac{16316775383}{10000000000}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, x, \frac{16316775383}{10000000000}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, x, \frac{-2134856267379707}{1000000000000000}\right)}, x, \frac{16316775383}{10000000000}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}}, x, \frac{-2134856267379707}{1000000000000000}\right), x, \frac{16316775383}{10000000000}\right) \]

      9. lower-fma.f64 99.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right)}, x, -2.134856267379707\right), x, 1.6316775383\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)} \]

   if 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.6%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\frac{\frac{27061}{4481}}{x}} - x\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 98.1

         \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - x\right) \cdot 0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_1 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.70711 x (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ (* x 0.27061) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_1 2.0)
     t_0
     (if (<= t_1 4.0)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(-0.70711, x, (4.2702753202410175 / x));
	double t_1 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(-0.70711, x, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_1 <= 2.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$1, 2.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. remove-double-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-70711}{100000}, x, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \]

   if 2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) \cdot x} + \frac{16316775383}{10000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, x, \frac{16316775383}{10000000000}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), x, \frac{16316775383}{10000000000}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) \cdot x + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, x, \frac{16316775383}{10000000000}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, x, \frac{-2134856267379707}{1000000000000000}\right)}, x, \frac{16316775383}{10000000000}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}}, x, \frac{-2134856267379707}{1000000000000000}\right), x, \frac{16316775383}{10000000000}\right) \]

      9. lower-fma.f64 99.6

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right)}, x, -2.134856267379707\right), x, 1.6316775383\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_1 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma -0.70711 x (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ (* x 0.27061) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_1 2.0)
     t_0
     (if (<= t_1 4.0)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(-0.70711, x, (4.2702753202410175 / x));
	double t_1 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_1 <= 2.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(-0.70711, x, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_1 <= 2.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.70711 * x + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$1, 2.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. remove-double-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{-70711}{100000} \cdot x + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-70711}{100000}, x, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      13. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      14. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-70711}{100000}, x, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \]

   if 2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) \cdot x} + \frac{16316775383}{10000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, x, \frac{16316775383}{10000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, x, \frac{16316775383}{10000000000}\right) \]

      6. lower-fma.f64 99.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right)}, x, 1.6316775383\right) \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* x 0.27061) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 2.0)
     (* -0.70711 x)
     (if (<= t_0 4.0)
       (fma (fma 1.3436228731669864 x -2.134856267379707) x 1.6316775383)
       (* -0.70711 x)))))

C

double code(double x) {
	double t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 4.0) {
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(-0.70711 * x);
	elseif (t_0 <= 4.0)
		tmp = fma(fma(1.3436228731669864, x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(1.3436228731669864 * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 97.5

         \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   if 2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) \cdot x} + \frac{16316775383}{10000000000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}, x, \frac{16316775383}{10000000000}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, x, \frac{16316775383}{10000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{134362287316698645903}{100000000000000000000} \cdot x + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, x, \frac{16316775383}{10000000000}\right) \]

      6. lower-fma.f64 99.5

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right)}, x, 1.6316775383\right) \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.3436228731669864, x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* x 0.27061) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 2.0)
     (* -0.70711 x)
     (if (<= t_0 4.0)
       (fma -2.134856267379707 x 1.6316775383)
       (* -0.70711 x)))))

C

double code(double x) {
	double t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 4.0) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(-0.70711 * x);
	elseif (t_0 <= 4.0)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], N[(-0.70711 * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 2 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 97.5

         \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   if 2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + \frac{-2134856267379707}{1000000000000000} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-2134856267379707}{1000000000000000} \cdot x + \frac{16316775383}{10000000000}} \]

      2. lower-fma.f64 99.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 2:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* x 0.27061) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -2.0)
     (* -0.70711 x)
     (if (<= t_0 4.0) 1.6316775383 (* -0.70711 x)))))

C

double code(double x) {
	double t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -2.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 4.0) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * 0.27061d0) + 2.30753d0) / ((((0.04481d0 * x) + 0.99229d0) * x) + 1.0d0)) - x
    if (t_0 <= (-2.0d0)) then
        tmp = (-0.70711d0) * x
    else if (t_0 <= 4.0d0) then
        tmp = 1.6316775383d0
    else
        tmp = (-0.70711d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -2.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 4.0) {
		tmp = 1.6316775383;
	} else {
		tmp = -0.70711 * x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x
	tmp = 0
	if t_0 <= -2.0:
		tmp = -0.70711 * x
	elif t_0 <= 4.0:
		tmp = 1.6316775383
	else:
		tmp = -0.70711 * x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = Float64(-0.70711 * x);
	elseif (t_0 <= 4.0)
		tmp = 1.6316775383;
	else
		tmp = Float64(-0.70711 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (((x * 0.27061) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = -0.70711 * x;
	elseif (t_0 <= 4.0)
		tmp = 1.6316775383;
	else
		tmp = -0.70711 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[t$95$0, 4.0], 1.6316775383, N[(-0.70711 * x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -2 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 98.1

         \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]

   if -2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{1.6316775383} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -2:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;\frac{x \cdot 0.27061 + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;-0.70711 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}, 0.70711, -0.70711 \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (/ (fma 0.27061 x 2.30753) (fma (fma 0.04481 x 0.99229) x 1.0))
  0.70711
  (* -0.70711 x)))

C

double code(double x) {
	return fma((fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)), 0.70711, (-0.70711 * x));
}

Julia

function code(x)
	return fma(Float64(fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)), 0.70711, Float64(-0.70711 * x))
end

Wolfram

code[x_] := N[(N[(N[(0.27061 * x + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * 0.70711 + N[(-0.70711 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   3. sub-neg N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   6. clear-num N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{1}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   7. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{70711}{100000}}}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   9. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}} \cdot \frac{70711}{100000}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   10. clear-num N/A

       \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}, \frac{70711}{100000}, \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}, 0.70711, -0.70711 \cdot x\right)} \]
5. Add Preprocessing

Alternative 11: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (fma 0.27061 x 2.30753) (fma (fma 0.04481 x 0.99229) x 1.0)) x)))

C

double code(double x) {
	return 0.70711 * ((fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x);
}

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x))
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(0.27061 * x + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000}} \]

   3. lower-*.f64 99.8

      \[\leadsto \color{blue}{\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \cdot 0.70711} \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   5. +-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   7. *-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   8. lower-fma.f64 99.8

      \[\leadsto \left(\frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \cdot 0.70711 \]

   9. lift-+.f64 N/A

      \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x\right) \cdot \frac{70711}{100000} \]

   10. +-commutative N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x\right) \cdot \frac{70711}{100000} \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x\right) \cdot \frac{70711}{100000} \]

   12. *-commutative N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x\right) \cdot \frac{70711}{100000} \]

   13. lower-fma.f64 99.8

       \[\leadsto \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x\right) \cdot 0.70711 \]

   14. lift-+.f64 N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   15. +-commutative N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   16. lift-*.f64 N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   17. *-commutative N/A

       \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   18. lower-fma.f64 99.8

       \[\leadsto \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x\right) \cdot 0.70711 \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711} \]

5. Final simplification 99.8%

   \[\leadsto 0.70711 \cdot \left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \]
6. Add Preprocessing

Alternative 12: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(0.99229, x, 1\right)} - x\right) \cdot 0.70711 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- (/ (fma x 0.27061 2.30753) (fma 0.99229 x 1.0)) x) 0.70711))

C

double code(double x) {
	return ((fma(x, 0.27061, 2.30753) / fma(0.99229, x, 1.0)) - x) * 0.70711;
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(x, 0.27061, 2.30753) / fma(0.99229, x, 1.0)) - x) * 0.70711)
end

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(0.99229 * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   3. flip-- N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}} \]

   4. clear-num N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{1}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + x}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \cdot x}}} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{0.70711}{\frac{1}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{70711}{100000}}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{1} \cdot \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right)} \]

   4. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{70711}{100000}} \cdot \left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000}} \]

   6. lower-*.f64 99.8

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711} \]

   7. lift-fma.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\frac{27061}{100000} \cdot x + \frac{230753}{100000}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right), x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   9. lower-fma.f64 99.8

      \[\leadsto \left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711 \]

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x\right) \cdot 0.70711} \]

7. Taylor expanded in x around 0

   \[\leadsto \left(\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000}}, x, 1\right)} - x\right) \cdot \frac{70711}{100000} \]
8. Step-by-step derivation

9. Applied rewrites 97.9%

   \[\leadsto \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{0.99229}, x, 1\right)} - x\right) \cdot 0.70711 \]
10. Add Preprocessing

Alternative 13: 49.5% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 1.6316775383 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.6316775383d0
end function

Java

public static double code(double x) {
	return 1.6316775383;
}

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

function tmp = code(x)
	tmp = 1.6316775383;
end

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{16316775383}{10000000000}} \]
4. Step-by-step derivation

5. Applied rewrites 48.3%

   \[\leadsto \color{blue}{1.6316775383} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))