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

Percentage Accurate: 99.9% → 99.9%

Time: 12.1s

Alternatives: 12

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. 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. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. metadata-eval N/A

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

   12. associate-*l/ N/A

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

   13. clear-num N/A

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

   14. lift-/.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}\right)\\ t_1 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;0.70711 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -3.0191289437\right), 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          x
          -0.70711
          (/ (fma x 0.1913510371 1.6316775383) (* x (fma x 0.04481 0.99229)))))
        (t_1
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_1 -5000.0)
     t_0
     (if (<= t_1 4.0)
       (*
        0.70711
        (fma
         x
         (fma x (fma x -1.7950336306565942 1.900161040244073) -3.0191289437)
         2.30753))
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (fma(x, 0.1913510371, 1.6316775383) / (x * fma(x, 0.04481, 0.99229))));
	double t_1 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = 0.70711 * fma(x, fma(x, fma(x, -1.7950336306565942, 1.900161040244073), -3.0191289437), 2.30753);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(fma(x, 0.1913510371, 1.6316775383) / Float64(x * fma(x, 0.04481, 0.99229))))
	t_1 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = Float64(0.70711 * fma(x, fma(x, fma(x, -1.7950336306565942, 1.900161040244073), -3.0191289437), 2.30753));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(N[(x * 0.1913510371 + 1.6316775383), $MachinePrecision] / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5000.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(0.70711 * N[(x * N[(x * N[(x * -1.7950336306565942 + 1.900161040244073), $MachinePrecision] + -3.0191289437), $MachinePrecision] + 2.30753), $MachinePrecision]), $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) < -5e3 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. 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. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.8

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\left(x \cdot \frac{27061}{100000}\right) \cdot \frac{70711}{100000} + \color{blue}{\frac{16316775383}{10000000000}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{x \cdot \left(\frac{27061}{100000} \cdot \frac{70711}{100000}\right)} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{x \cdot \color{blue}{\frac{1913510371}{10000000000}} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{x \cdot \color{blue}{\left(\frac{70711}{100000} \cdot \frac{27061}{100000}\right)} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\mathsf{fma}\left(x, \frac{70711}{100000} \cdot \frac{27061}{100000}, \frac{16316775383}{10000000000}\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      12. metadata-eval 99.8

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{\color{blue}{{x}^{2} \cdot \left(\frac{4481}{100000} + \frac{99229}{100000} \cdot \frac{1}{x}\right)}}\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{4481}{100000} + \frac{99229}{100000} \cdot \frac{1}{x}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{4481}{100000} + \frac{99229}{100000} \cdot \frac{1}{x}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{\color{blue}{x \cdot \left(x \cdot \left(\frac{4481}{100000} + \frac{99229}{100000} \cdot \frac{1}{x}\right)\right)}}\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{x \cdot \color{blue}{\left(\frac{4481}{100000} \cdot x + \left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{x \cdot \left(\color{blue}{x \cdot \frac{4481}{100000}} + \left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right)}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{4481}{100000}, \left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right)}}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{x \cdot \mathsf{fma}\left(x, \frac{4481}{100000}, \color{blue}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot x\right)}\right)}\right) \]

      8. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{10000000000}, \frac{16316775383}{10000000000}\right)}{x \cdot \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000} \cdot \color{blue}{1}\right)}\right) \]

      9. metadata-eval 99.4

         \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{x \cdot \mathsf{fma}\left(x, 0.04481, \color{blue}{0.99229}\right)}\right) \]

   9. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}}\right) \]

   if -5e3 < (-.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

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}, \frac{230753}{100000}\right)} - x\right) \]

      3. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right)}, \frac{230753}{100000}\right) - x\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \color{blue}{\frac{-20191289437}{10000000000}}, \frac{230753}{100000}\right) - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, \frac{-20191289437}{10000000000}\right)}, \frac{230753}{100000}\right) - x\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, \frac{-20191289437}{10000000000}\right), \frac{230753}{100000}\right) - x\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-179503363065659419717}{100000000000000000000}} + \frac{1900161040244073}{1000000000000000}, \frac{-20191289437}{10000000000}\right), \frac{230753}{100000}\right) - x\right) \]

      8. lower-fma.f64 99.3

         \[\leadsto 0.70711 \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right)}, -2.0191289437\right), 2.30753\right) - x\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -2.0191289437\right), 2.30753\right)} - x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}, \frac{230753}{100000}\right)} \]

      3. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right)}, \frac{230753}{100000}\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \color{blue}{\frac{-30191289437}{10000000000}}, \frac{230753}{100000}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, \frac{-30191289437}{10000000000}\right)}, \frac{230753}{100000}\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, \frac{-30191289437}{10000000000}\right), \frac{230753}{100000}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-179503363065659419717}{100000000000000000000}} + \frac{1900161040244073}{1000000000000000}, \frac{-30191289437}{10000000000}\right), \frac{230753}{100000}\right) \]

      8. lower-fma.f64 99.3

         \[\leadsto 0.70711 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right)}, -3.0191289437\right), 2.30753\right) \]

   8. Applied rewrites 99.3%

      \[\leadsto 0.70711 \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -3.0191289437\right), 2.30753\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 4.2702753202410175, -58.14938538768042\right)}{x \cdot x}\right)\\ t_1 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;0.70711 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -3.0191289437\right), 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          x
          -0.70711
          (/ (fma x 4.2702753202410175 -58.14938538768042) (* x x))))
        (t_1
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_1 -5000.0)
     t_0
     (if (<= t_1 4.0)
       (*
        0.70711
        (fma
         x
         (fma x (fma x -1.7950336306565942 1.900161040244073) -3.0191289437)
         2.30753))
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (fma(x, 4.2702753202410175, -58.14938538768042) / (x * x)));
	double t_1 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = 0.70711 * fma(x, fma(x, fma(x, -1.7950336306565942, 1.900161040244073), -3.0191289437), 2.30753);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(fma(x, 4.2702753202410175, -58.14938538768042) / Float64(x * x)))
	t_1 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = Float64(0.70711 * fma(x, fma(x, fma(x, -1.7950336306565942, 1.900161040244073), -3.0191289437), 2.30753));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(N[(x * 4.2702753202410175 + -58.14938538768042), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5000.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(0.70711 * N[(x * N[(x * N[(x * -1.7950336306565942 + 1.900161040244073), $MachinePrecision] + -3.0191289437), $MachinePrecision] + 2.30753), $MachinePrecision]), $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) < -5e3 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. 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. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. neg-mul-1 N/A

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

      7. associate-*r* N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.8

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\left(x \cdot \frac{27061}{100000}\right) \cdot \frac{70711}{100000} + \color{blue}{\frac{16316775383}{10000000000}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{x \cdot \left(\frac{27061}{100000} \cdot \frac{70711}{100000}\right)} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{x \cdot \color{blue}{\frac{1913510371}{10000000000}} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{x \cdot \color{blue}{\left(\frac{70711}{100000} \cdot \frac{27061}{100000}\right)} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\mathsf{fma}\left(x, \frac{70711}{100000} \cdot \frac{27061}{100000}, \frac{16316775383}{10000000000}\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

      12. metadata-eval 99.8

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000} - \frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}}\right) \]
   8. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000}}{x} - \frac{\frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{1913510371}{448100000} \cdot 1}}{x} - \frac{\frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000}}{x} \cdot 1} - \frac{\frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000}}{x} \cdot \color{blue}{\frac{x}{x}} - \frac{\frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000} \cdot x}{x \cdot x}} - \frac{\frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} \cdot x}{\color{blue}{{x}^{2}}} - \frac{\frac{3648757816023}{62748003125} \cdot \frac{1}{x}}{x}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} \cdot x}{{x}^{2}} - \frac{\color{blue}{\frac{\frac{3648757816023}{62748003125} \cdot 1}{x}}}{x}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} \cdot x}{{x}^{2}} - \frac{\frac{\color{blue}{\frac{3648757816023}{62748003125}}}{x}}{x}\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} \cdot x}{{x}^{2}} - \color{blue}{\frac{\frac{3648757816023}{62748003125}}{x \cdot x}}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{1913510371}{448100000} \cdot x}{{x}^{2}} - \frac{\frac{3648757816023}{62748003125}}{\color{blue}{{x}^{2}}}\right) \]

      11. div-sub N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000} \cdot x - \frac{3648757816023}{62748003125}}{{x}^{2}}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{1913510371}{448100000} \cdot x - \frac{3648757816023}{62748003125}}{{x}^{2}}}\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{1913510371}{448100000} \cdot x + \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right)}}{{x}^{2}}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{x \cdot \frac{1913510371}{448100000}} + \left(\mathsf{neg}\left(\frac{3648757816023}{62748003125}\right)\right)}{{x}^{2}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{x \cdot \frac{1913510371}{448100000} + \color{blue}{\frac{-3648757816023}{62748003125}}}{{x}^{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\mathsf{fma}\left(x, \frac{1913510371}{448100000}, \frac{-3648757816023}{62748003125}\right)}}{{x}^{2}}\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\mathsf{fma}\left(x, \frac{1913510371}{448100000}, \frac{-3648757816023}{62748003125}\right)}{\color{blue}{x \cdot x}}\right) \]

      18. lower-*.f64 99.3

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 4.2702753202410175, -58.14938538768042\right)}{\color{blue}{x \cdot x}}\right) \]

   9. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{\mathsf{fma}\left(x, 4.2702753202410175, -58.14938538768042\right)}{x \cdot x}}\right) \]

   if -5e3 < (-.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

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}, \frac{230753}{100000}\right)} - x\right) \]

      3. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right)}, \frac{230753}{100000}\right) - x\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \color{blue}{\frac{-20191289437}{10000000000}}, \frac{230753}{100000}\right) - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, \frac{-20191289437}{10000000000}\right)}, \frac{230753}{100000}\right) - x\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, \frac{-20191289437}{10000000000}\right), \frac{230753}{100000}\right) - x\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-179503363065659419717}{100000000000000000000}} + \frac{1900161040244073}{1000000000000000}, \frac{-20191289437}{10000000000}\right), \frac{230753}{100000}\right) - x\right) \]

      8. lower-fma.f64 99.3

         \[\leadsto 0.70711 \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right)}, -2.0191289437\right), 2.30753\right) - x\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -2.0191289437\right), 2.30753\right)} - x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}, \frac{230753}{100000}\right)} \]

      3. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right)}, \frac{230753}{100000}\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \color{blue}{\frac{-30191289437}{10000000000}}, \frac{230753}{100000}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, \frac{-30191289437}{10000000000}\right)}, \frac{230753}{100000}\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, \frac{-30191289437}{10000000000}\right), \frac{230753}{100000}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-179503363065659419717}{100000000000000000000}} + \frac{1900161040244073}{1000000000000000}, \frac{-30191289437}{10000000000}\right), \frac{230753}{100000}\right) \]

      8. lower-fma.f64 99.3

         \[\leadsto 0.70711 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right)}, -3.0191289437\right), 2.30753\right) \]

   8. Applied rewrites 99.3%

      \[\leadsto 0.70711 \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -3.0191289437\right), 2.30753\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;0.70711 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -3.0191289437\right), 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_1 -5000.0)
     t_0
     (if (<= t_1 4.0)
       (*
        0.70711
        (fma
         x
         (fma x (fma x -1.7950336306565942 1.900161040244073) -3.0191289437)
         2.30753))
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double t_1 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = 0.70711 * fma(x, fma(x, fma(x, -1.7950336306565942, 1.900161040244073), -3.0191289437), 2.30753);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = Float64(0.70711 * fma(x, fma(x, fma(x, -1.7950336306565942, 1.900161040244073), -3.0191289437), 2.30753));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5000.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(0.70711 * N[(x * N[(x * N[(x * -1.7950336306565942 + 1.900161040244073), $MachinePrecision] + -3.0191289437), $MachinePrecision] + 2.30753), $MachinePrecision]), $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) < -5e3 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. *-commutative N/A

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

      6. remove-double-neg N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 99.2%

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

   if -5e3 < (-.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

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}, \frac{230753}{100000}\right)} - x\right) \]

      3. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right)}, \frac{230753}{100000}\right) - x\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \color{blue}{\frac{-20191289437}{10000000000}}, \frac{230753}{100000}\right) - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, \frac{-20191289437}{10000000000}\right)}, \frac{230753}{100000}\right) - x\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, \frac{-20191289437}{10000000000}\right), \frac{230753}{100000}\right) - x\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-179503363065659419717}{100000000000000000000}} + \frac{1900161040244073}{1000000000000000}, \frac{-20191289437}{10000000000}\right), \frac{230753}{100000}\right) - x\right) \]

      8. lower-fma.f64 99.3

         \[\leadsto 0.70711 \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right)}, -2.0191289437\right), 2.30753\right) - x\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -2.0191289437\right), 2.30753\right)} - x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}, \frac{230753}{100000}\right)} \]

      3. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right)}, \frac{230753}{100000}\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \color{blue}{\frac{-30191289437}{10000000000}}, \frac{230753}{100000}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, \frac{-30191289437}{10000000000}\right)}, \frac{230753}{100000}\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, \frac{-30191289437}{10000000000}\right), \frac{230753}{100000}\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-179503363065659419717}{100000000000000000000}} + \frac{1900161040244073}{1000000000000000}, \frac{-30191289437}{10000000000}\right), \frac{230753}{100000}\right) \]

      8. lower-fma.f64 99.3

         \[\leadsto 0.70711 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right)}, -3.0191289437\right), 2.30753\right) \]

   8. Applied rewrites 99.3%

      \[\leadsto 0.70711 \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -3.0191289437\right), 2.30753\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.2692862305735844, 1.3436228731669864\right), -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_1 -5000.0)
     t_0
     (if (<= t_1 4.0)
       (fma
        x
        (fma
         x
         (fma x -1.2692862305735844 1.3436228731669864)
         -2.134856267379707)
        1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double t_1 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = fma(x, fma(x, fma(x, -1.2692862305735844, 1.3436228731669864), -2.134856267379707), 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = fma(x, fma(x, fma(x, -1.2692862305735844, 1.3436228731669864), -2.134856267379707), 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5000.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(x * N[(x * N[(x * -1.2692862305735844 + 1.3436228731669864), $MachinePrecision] + -2.134856267379707), $MachinePrecision] + 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) < -5e3 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. *-commutative N/A

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

      6. remove-double-neg N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 99.2%

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

   if -5e3 < (-.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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 6: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ t_1 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_1 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x)))
        (t_1
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_1 -5000.0)
     t_0
     (if (<= t_1 4.0)
       (fma x (fma x 1.3436228731669864 -2.134856267379707) 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double t_1 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0;
	} else if (t_1 <= 4.0) {
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	t_1 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = t_0;
	elseif (t_1 <= 4.0)
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -5000.0], t$95$0, If[LessEqual[t$95$1, 4.0], N[(x * N[(x * 1.3436228731669864 + -2.134856267379707), $MachinePrecision] + 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) < -5e3 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. *-commutative N/A

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

      6. remove-double-neg N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 99.2%

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

   if -5e3 < (-.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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 7: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -5000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_0 -5000.0)
     (* x -0.70711)
     (if (<= t_0 4.0)
       (fma x (fma x 1.3436228731669864 -2.134856267379707) 1.6316775383)
       (* x -0.70711)))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_0 <= -5000.0) {
		tmp = x * -0.70711;
	} else if (t_0 <= 4.0) {
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_0 <= -5000.0)
		tmp = Float64(x * -0.70711);
	elseif (t_0 <= 4.0)
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -5000.0], N[(x * -0.70711), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(x * N[(x * 1.3436228731669864 + -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $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) < -5e3 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. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} \]

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -5e3 < (-.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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 8: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -5000:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_0 -5000.0)
     (* x -0.70711)
     (if (<= t_0 4.0) 1.6316775383 (* x -0.70711)))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_0 <= -5000.0) {
		tmp = x * -0.70711;
	} else if (t_0 <= 4.0) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
    if (t_0 <= (-5000.0d0)) then
        tmp = x * (-0.70711d0)
    else if (t_0 <= 4.0d0) then
        tmp = 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_0 <= -5000.0) {
		tmp = x * -0.70711;
	} else if (t_0 <= 4.0) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x
	tmp = 0
	if t_0 <= -5000.0:
		tmp = x * -0.70711
	elif t_0 <= 4.0:
		tmp = 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_0 <= -5000.0)
		tmp = Float64(x * -0.70711);
	elseif (t_0 <= 4.0)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	tmp = 0.0;
	if (t_0 <= -5000.0)
		tmp = x * -0.70711;
	elseif (t_0 <= 4.0)
		tmp = 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -5000.0], N[(x * -0.70711), $MachinePrecision], If[LessEqual[t$95$0, 4.0], 1.6316775383, N[(x * -0.70711), $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) < -5e3 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. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} \]

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -5e3 < (-.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 97.2%

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

Alternative 9: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  -0.70711
  (/ (fma x 0.1913510371 1.6316775383) (fma x (fma x 0.04481 0.99229) 1.0))))

C

double code(double x) {
	return fma(x, -0.70711, (fma(x, 0.1913510371, 1.6316775383) / fma(x, fma(x, 0.04481, 0.99229), 1.0)));
}

Julia

function code(x)
	return fma(x, -0.70711, Float64(fma(x, 0.1913510371, 1.6316775383) / fma(x, fma(x, 0.04481, 0.99229), 1.0)))
end

Wolfram

code[x_] := N[(x * -0.70711 + N[(N[(x * 0.1913510371 + 1.6316775383), $MachinePrecision] / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. 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. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. metadata-eval N/A

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

   12. associate-*l/ N/A

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

   13. clear-num N/A

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

   14. lift-/.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 99.9

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\left(x \cdot \frac{27061}{100000}\right) \cdot \frac{70711}{100000} + \color{blue}{\frac{16316775383}{10000000000}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{x \cdot \left(\frac{27061}{100000} \cdot \frac{70711}{100000}\right)} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{x \cdot \color{blue}{\frac{1913510371}{10000000000}} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{x \cdot \color{blue}{\left(\frac{70711}{100000} \cdot \frac{27061}{100000}\right)} + \frac{16316775383}{10000000000}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\mathsf{fma}\left(x, \frac{70711}{100000} \cdot \frac{27061}{100000}, \frac{16316775383}{10000000000}\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), 1\right)}\right) \]

   12. metadata-eval 99.9

       \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right) \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
7. Add Preprocessing

Alternative 10: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;0.70711 \cdot \mathsf{fma}\left(x, -3.0191289437, 2.30753\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 1.15) (* 0.70711 (fma x -3.0191289437 2.30753)) (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 0.70711 * fma(x, -3.0191289437, 2.30753);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = Float64(0.70711 * fma(x, -3.0191289437, 2.30753));
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], N[(0.70711 * N[(x * -3.0191289437 + 2.30753), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} \]

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   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

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}, \frac{230753}{100000}\right)} - x\right) \]

      3. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right)}, \frac{230753}{100000}\right) - x\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) + \color{blue}{\frac{-20191289437}{10000000000}}, \frac{230753}{100000}\right) - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x, \frac{-20191289437}{10000000000}\right)}, \frac{230753}{100000}\right) - x\right) \]

      6. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}}, \frac{-20191289437}{10000000000}\right), \frac{230753}{100000}\right) - x\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-179503363065659419717}{100000000000000000000}} + \frac{1900161040244073}{1000000000000000}, \frac{-20191289437}{10000000000}\right), \frac{230753}{100000}\right) - x\right) \]

      8. lower-fma.f64 99.3

         \[\leadsto 0.70711 \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right)}, -2.0191289437\right), 2.30753\right) - x\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.7950336306565942, 1.900161040244073\right), -2.0191289437\right), 2.30753\right)} - x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{x \cdot \frac{-30191289437}{10000000000}} + \frac{230753}{100000}\right) \]

      3. lower-fma.f64 98.3

         \[\leadsto 0.70711 \cdot \color{blue}{\mathsf{fma}\left(x, -3.0191289437, 2.30753\right)} \]

   8. Applied rewrites 98.3%

      \[\leadsto 0.70711 \cdot \color{blue}{\mathsf{fma}\left(x, -3.0191289437, 2.30753\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 98.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(-2.134856267379707, x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 1.15) (fma -2.134856267379707 x 1.6316775383) (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = fma(-2.134856267379707, x, 1.6316775383);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], N[(-2.134856267379707 * x + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} \]

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   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 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 12: 51.0% 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.0%

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

Reproduce

herbie shell --seed 2024238 
(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)))