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

Percentage Accurate: 99.9% → 99.9%

Time: 12.1s

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. lft-mult-inverse N/A

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

   5. associate-*l* N/A

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

   6. distribute-rgt-in N/A

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

   7. associate-*l* N/A

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

   8. unpow2 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{{x}^{2}} \cdot \left(\frac{4481}{100000} + \frac{99229}{100000} \cdot \frac{1}{x}\right) + 1} - x\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{4481}{100000} + \frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}} + 1} - x\right) \]

   10. *-commutative N/A

       \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{{x}^{2} \cdot \left(\frac{4481}{100000} + \frac{99229}{100000} \cdot \frac{1}{x}\right)} + 1} - x\right) \]

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-fma.f64 N/A

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

   14. distribute-rgt-in N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. associate-*l* N/A

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

   18. lft-mult-inverse N/A

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

   19. metadata-eval 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 99.8

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

8. Simplified 99.8%

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

Alternative 2: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;x \cdot -2.134856267379707\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707\\ \end{array} \end{array} \]

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 -0.1)
     (* x -2.134856267379707)
     (if (<= t_0 5.0) 1.6316775383 (* x -2.134856267379707)))))

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 <= -0.1) {
		tmp = x * -2.134856267379707;
	} else if (t_0 <= 5.0) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -2.134856267379707;
	}
	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 <= (-0.1d0)) then
        tmp = x * (-2.134856267379707d0)
    else if (t_0 <= 5.0d0) then
        tmp = 1.6316775383d0
    else
        tmp = x * (-2.134856267379707d0)
    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 <= -0.1) {
		tmp = x * -2.134856267379707;
	} else if (t_0 <= 5.0) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -2.134856267379707;
	}
	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 <= -0.1:
		tmp = x * -2.134856267379707
	elif t_0 <= 5.0:
		tmp = 1.6316775383
	else:
		tmp = x * -2.134856267379707
	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 <= -0.1)
		tmp = Float64(x * -2.134856267379707);
	elseif (t_0 <= 5.0)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -2.134856267379707);
	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 <= -0.1)
		tmp = x * -2.134856267379707;
	elseif (t_0 <= 5.0)
		tmp = 1.6316775383;
	else
		tmp = x * -2.134856267379707;
	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, -0.1], N[(x * -2.134856267379707), $MachinePrecision], If[LessEqual[t$95$0, 5.0], 1.6316775383, N[(x * -2.134856267379707), $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) < -0.10000000000000001 or 5 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 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 17.7

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

   5. Simplified 17.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2134856267379707}{1000000000000000} \cdot x} \]
   7. Step-by-step derivation

      1. lower-*.f64 17.7

         \[\leadsto \color{blue}{-2.134856267379707 \cdot x} \]

   8. Simplified 17.7%

      \[\leadsto \color{blue}{-2.134856267379707 \cdot x} \]

   if -0.10000000000000001 < (-.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) < 5

   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. Simplified 97.4%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -0.1:\\ \;\;\;\;x \cdot -2.134856267379707\\ \mathbf{elif}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 5:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot \color{blue}{{x}^{2}}\right) + 1} - x\right) \]

   6. associate-*l* N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}} + 1} - x\right) \]

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-*l* N/A

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

   12. lft-mult-inverse N/A

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

   13. metadata-eval 98.2

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

5. Simplified 98.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 98.2

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

8. Simplified 98.2%

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

Alternative 4: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.2692862305735844, 1.3436228731669864\right), -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -1.05)
     t_0
     (if (<= x 2.5)
       (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 tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 2.5) {
		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))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 2.5)
		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]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 2.5], 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 x < -1.05000000000000004 or 2.5 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   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. Simplified 99.0%

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

   if -1.05000000000000004 < x < 2.5

   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 98.5

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

   5. Simplified 98.5%

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

Alternative 5: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -1.05)
     t_0
     (if (<= x 1.55)
       (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 tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.55) {
		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))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.55)
		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]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 1.55], N[(x * N[(x * 1.3436228731669864 + -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.55000000000000004 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   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. Simplified 99.0%

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

   if -1.05000000000000004 < x < 1.55000000000000004

   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 98.1

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

   5. Simplified 98.1%

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

Alternative 6: 98.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot \color{blue}{{x}^{2}}\right) + 1} - x\right) \]

   6. associate-*l* N/A

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}} + 1} - x\right) \]

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-*l* N/A

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

   12. lft-mult-inverse N/A

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

   13. metadata-eval 98.2

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

5. Simplified 98.2%

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

6. Taylor expanded in x around 0

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

8. Simplified 98.0%

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

Alternative 7: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 1.2)
     (fma x (fma x 1.3436228731669864 -2.134856267379707) 1.6316775383)
     (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.2) {
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 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 <= 1.2)
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 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, 1.2], 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 x < -1.05000000000000004 or 1.19999999999999996 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   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 98.8

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

   5. Simplified 98.8%

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

Alternative 8: 98.7% 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 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

   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 98.6

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

   5. Simplified 98.6%

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

Alternative 9: 98.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) (* x -0.70711) (if (<= x 1.2) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.2) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.2d0) then
        tmp = 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.2) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x * -0.70711
	elif x <= 1.2:
		tmp = 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 <= 1.2)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x * -0.70711;
	elseif (x <= 1.2)
		tmp = 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.2], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.19999999999999996 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.2

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

   5. Simplified 98.2%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   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. Simplified 97.4%

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

Alternative 10: 50.6% accurate, 44.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

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

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.5%

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

Reproduce

herbie shell --seed 2024208 
(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)))