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

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. metadata-eval N/A

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

   9. associate-*l/ N/A

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

   10. clear-num N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 99.8

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x\\ \mathbf{if}\;t\_0 \leq -1000000000:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(6.039053782637804, x, -82.23527511657367\right)}{x \cdot x} - x\right) \cdot 0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ (* 0.27061 x) 2.30753) (+ (* (+ (* 0.04481 x) 0.99229) x) 1.0))
          x)))
   (if (<= t_0 -1000000000.0)
     (* -0.70711 x)
     (if (<= t_0 4.0)
       (fma
        (fma
         (fma -1.2692862305735844 x 1.3436228731669864)
         x
         -2.134856267379707)
        x
        1.6316775383)
       (*
        (- (/ (fma 6.039053782637804 x -82.23527511657367) (* x x)) x)
        0.70711)))))

C

double code(double x) {
	double t_0 = (((0.27061 * x) + 2.30753) / ((((0.04481 * x) + 0.99229) * x) + 1.0)) - x;
	double tmp;
	if (t_0 <= -1000000000.0) {
		tmp = -0.70711 * x;
	} else if (t_0 <= 4.0) {
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	} else {
		tmp = ((fma(6.039053782637804, x, -82.23527511657367) / (x * x)) - x) * 0.70711;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(Float64(Float64(Float64(0.04481 * x) + 0.99229) * x) + 1.0)) - x)
	tmp = 0.0
	if (t_0 <= -1000000000.0)
		tmp = Float64(-0.70711 * x);
	elseif (t_0 <= 4.0)
		tmp = fma(fma(fma(-1.2692862305735844, x, 1.3436228731669864), x, -2.134856267379707), x, 1.6316775383);
	else
		tmp = Float64(Float64(Float64(fma(6.039053782637804, x, -82.23527511657367) / Float64(x * x)) - x) * 0.70711);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(N[(N[(0.04481 * x), $MachinePrecision] + 0.99229), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000000.0], N[(-0.70711 * x), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(N[(-1.2692862305735844 * x + 1.3436228731669864), $MachinePrecision] * x + -2.134856267379707), $MachinePrecision] * x + 1.6316775383), $MachinePrecision], N[(N[(N[(N[(6.039053782637804 * x + -82.23527511657367), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] * 0.70711), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -1e9

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -1e9 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

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

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 99.8

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}} - x\right) \cdot \frac{70711}{100000} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \left(\color{blue}{\left(\frac{\frac{27061}{4481}}{x} - \frac{\frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}\right)} - x\right) \cdot \frac{70711}{100000} \]

      2. sub-neg N/A

         \[\leadsto \left(\color{blue}{\left(\frac{\frac{27061}{4481}}{x} + \left(\mathsf{neg}\left(\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}\right)\right)\right)} - x\right) \cdot \frac{70711}{100000} \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}\right)\right) + \frac{\frac{27061}{4481}}{x}\right)} - x\right) \cdot \frac{70711}{100000} \]

      4. neg-sub0 N/A

         \[\leadsto \left(\left(\color{blue}{\left(0 - \frac{\frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}\right)} + \frac{\frac{27061}{4481}}{x}\right) - x\right) \cdot \frac{70711}{100000} \]

      5. associate--r- N/A

         \[\leadsto \left(\color{blue}{\left(0 - \left(\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - \frac{\frac{27061}{4481}}{x}\right)\right)} - x\right) \cdot \frac{70711}{100000} \]

      6. div-sub N/A

         \[\leadsto \left(\left(0 - \color{blue}{\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x} - \frac{27061}{4481}}{x}}\right) - x\right) \cdot \frac{70711}{100000} \]

      7. neg-sub0 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x} - \frac{27061}{4481}}{x}\right)\right)} - x\right) \cdot \frac{70711}{100000} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x} - \frac{27061}{4481}}{\mathsf{neg}\left(x\right)}} - x\right) \cdot \frac{70711}{100000} \]

      9. mul-1-neg N/A

         \[\leadsto \left(\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x} - \frac{27061}{4481}}{\color{blue}{-1 \cdot x}} - x\right) \cdot \frac{70711}{100000} \]

      10. associate-/r* N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x} - \frac{27061}{4481}}{-1}}{x}} - x\right) \cdot \frac{70711}{100000} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{\frac{1651231776}{20079361} \cdot \frac{1}{x} - \frac{27061}{4481}}{-1}}{x}} - x\right) \cdot \frac{70711}{100000} \]

   7. Applied rewrites 99.2%

      \[\leadsto \left(\color{blue}{\frac{\frac{-82.23527511657367}{x} - -6.039053782637804}{x}} - x\right) \cdot 0.70711 \]

   8. Taylor expanded in x around inf

      \[\leadsto \left(\color{blue}{\frac{\frac{27061}{4481} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x}} - x\right) \cdot \frac{70711}{100000} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{27061}{4481} \cdot 1} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right) \cdot \frac{70711}{100000} \]

      2. *-inverses N/A

         \[\leadsto \left(\frac{\frac{27061}{4481} \cdot \color{blue}{\frac{x}{x}} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right) \cdot \frac{70711}{100000} \]

      3. associate-/l* N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{\frac{27061}{4481} \cdot x}{x}} - \frac{1651231776}{20079361} \cdot \frac{1}{x}}{x} - x\right) \cdot \frac{70711}{100000} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{\frac{\frac{27061}{4481} \cdot x}{x} - \color{blue}{\frac{\frac{1651231776}{20079361} \cdot 1}{x}}}{x} - x\right) \cdot \frac{70711}{100000} \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{\frac{\frac{27061}{4481} \cdot x}{x} - \frac{\color{blue}{\frac{1651231776}{20079361}}}{x}}{x} - x\right) \cdot \frac{70711}{100000} \]

      6. div-sub N/A

         \[\leadsto \left(\frac{\color{blue}{\frac{\frac{27061}{4481} \cdot x - \frac{1651231776}{20079361}}{x}}}{x} - x\right) \cdot \frac{70711}{100000} \]

      7. associate-/r* N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{27061}{4481} \cdot x - \frac{1651231776}{20079361}}{x \cdot x}} - x\right) \cdot \frac{70711}{100000} \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{\frac{27061}{4481} \cdot x - \frac{1651231776}{20079361}}{\color{blue}{{x}^{2}}} - x\right) \cdot \frac{70711}{100000} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{27061}{4481} \cdot x - \frac{1651231776}{20079361}}{{x}^{2}}} - x\right) \cdot \frac{70711}{100000} \]

      10. sub-neg N/A

          \[\leadsto \left(\frac{\color{blue}{\frac{27061}{4481} \cdot x + \left(\mathsf{neg}\left(\frac{1651231776}{20079361}\right)\right)}}{{x}^{2}} - x\right) \cdot \frac{70711}{100000} \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{\frac{27061}{4481} \cdot x + \color{blue}{\frac{-1651231776}{20079361}}}{{x}^{2}} - x\right) \cdot \frac{70711}{100000} \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(\frac{\color{blue}{\mathsf{fma}\left(\frac{27061}{4481}, x, \frac{-1651231776}{20079361}\right)}}{{x}^{2}} - x\right) \cdot \frac{70711}{100000} \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{27061}{4481}, x, \frac{-1651231776}{20079361}\right)}{\color{blue}{x \cdot x}} - x\right) \cdot \frac{70711}{100000} \]

      14. lower-*.f64 98.8

          \[\leadsto \left(\frac{\mathsf{fma}\left(6.039053782637804, x, -82.23527511657367\right)}{\color{blue}{x \cdot x}} - x\right) \cdot 0.70711 \]

   10. Applied rewrites 98.8%

       \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(6.039053782637804, x, -82.23527511657367\right)}{x \cdot x}} - x\right) \cdot 0.70711 \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq -1000000000:\\ \;\;\;\;-0.70711 \cdot x\\ \mathbf{elif}\;\frac{0.27061 \cdot x + 2.30753}{\left(0.04481 \cdot x + 0.99229\right) \cdot x + 1} - x \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.2692862305735844, x, 1.3436228731669864\right), x, -2.134856267379707\right), x, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(6.039053782637804, x, -82.23527511657367\right)}{x \cdot x} - x\right) \cdot 0.70711\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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)))