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

Percentage Accurate: 99.8% → 99.9%

Time: 10.2s

Alternatives: 9

Speedup: 1.0×

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (-0.70711d0)) - (((x * 0.1913510371d0) + 1.6316775383d0) / ((-1.0d0) - (x * ((x * 0.04481d0) + 0.99229d0))))
end function

Java

public static double code(double x) {
	return (x * -0.70711) - (((x * 0.1913510371) + 1.6316775383) / (-1.0 - (x * ((x * 0.04481) + 0.99229))));
}

Python

def code(x):
	return (x * -0.70711) - (((x * 0.1913510371) + 1.6316775383) / (-1.0 - (x * ((x * 0.04481) + 0.99229))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * -0.70711) - (((x * 0.1913510371) + 1.6316775383) / (-1.0 - (x * ((x * 0.04481) + 0.99229))));
end

Wolfram

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

   1. sub-neg 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-rgt-in 99.9%

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

   4. *-commutative 99.9%

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

   5. neg-mul-1 99.9%

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

   6. associate-*r* 99.9%

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

   7. *-commutative 99.9%

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

   8. fma-def 99.9%

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

   9. metadata-eval 99.9%

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

   10. associate-*l/ 99.9%

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

   11. *-commutative 99.9%

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

   12. +-commutative 99.9%

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

   13. distribute-lft-in 99.9%

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

   14. *-commutative 99.9%

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

   15. associate-*r* 99.9%

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

   16. *-commutative 99.9%

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

   17. fma-def 99.9%

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

   18. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

   19. metadata-eval 99.9%

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

   20. +-commutative 99.9%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

   2. frac-2neg 99.9%

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

   3. fma-udef 99.9%

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

   4. fma-udef 99.9%

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

   5. +-commutative 99.9%

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

   6. +-commutative 99.9%

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

   7. distribute-frac-neg 99.9%

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

   8. unsub-neg 99.9%

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

   9. neg-sub0 99.9%

      \[\leadsto x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{0 - \left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}} \]

   10. associate--r+ 99.9%

       \[\leadsto x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\left(0 - 1\right) - x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \]

   11. metadata-eval 99.9%

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

   12. +-commutative 99.9%

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

   13. fma-udef 99.9%

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

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}} \]
6. Step-by-step derivation

   1. fma-udef 99.9%

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

7. Applied egg-rr 99.9%

   \[\leadsto x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{-1 - x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)}} \]
8. Step-by-step derivation

   1. fma-udef 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

    \[\leadsto x \cdot -0.70711 - \frac{x \cdot 0.1913510371 + 1.6316775383}{-1 - x \cdot \left(x \cdot 0.04481 + 0.99229\right)} \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * 0.04481), $MachinePrecision] + 0.99229), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $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. Final simplification 99.9%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 3.55:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 3.55)
     (+ 1.6316775383 (* x -2.134856267379707))
     (* 0.70711 (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 3.55) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.06d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 3.55d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = 0.70711d0 * (((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 3.55) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 3.55:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.06)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 3.55)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.06)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 3.55)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.06], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.55], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0600000000000001

   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. Taylor expanded in x around inf 99.8%

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

   if -1.0600000000000001 < x < 3.5499999999999998

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. *-commutative 99.9%

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

      5. neg-mul-1 99.9%

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

      6. associate-*r* 99.9%

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

      7. *-commutative 99.9%

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

      8. fma-def 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*l/ 99.9%

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

      11. *-commutative 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-lft-in 99.9%

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

      14. *-commutative 99.9%

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

      15. associate-*r* 99.9%

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

      16. *-commutative 99.9%

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

      17. fma-def 99.9%

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

      18. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      19. metadata-eval 99.9%

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

      20. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto 1.6316775383 + \color{blue}{x \cdot -2.134856267379707} \]

   6. Simplified 99.2%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot -2.134856267379707} \]

   if 3.5499999999999998 < 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. Taylor expanded in x around inf 98.3%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(6.039053782637804 \cdot \frac{1}{x} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right)} - x\right) \]
   3. Step-by-step derivation

      1. associate-*r/ 98.3%

         \[\leadsto 0.70711 \cdot \left(\left(\color{blue}{\frac{6.039053782637804 \cdot 1}{x}} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right) - x\right) \]

      2. metadata-eval 98.3%

         \[\leadsto 0.70711 \cdot \left(\left(\frac{\color{blue}{6.039053782637804}}{x} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right) - x\right) \]

      3. associate-*r/ 98.3%

         \[\leadsto 0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} - \color{blue}{\frac{82.23527511657367 \cdot 1}{{x}^{2}}}\right) - x\right) \]

      4. metadata-eval 98.3%

         \[\leadsto 0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} - \frac{\color{blue}{82.23527511657367}}{{x}^{2}}\right) - x\right) \]

      5. unpow2 98.3%

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

   4. Simplified 98.3%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right)} - x\right) \]
   5. Step-by-step derivation

      1. associate-/r* 98.3%

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

      2. sub-div 98.3%

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

   6. Applied egg-rr 98.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 3.55:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $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. Taylor expanded in x around 0 98.8%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} - x\right) \]
3. Step-by-step derivation

   1. *-commutative 98.8%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 5: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.06) (not (<= x 2.8)))
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (+ 1.6316775383 (* x -2.134856267379707))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.06) || !(x <= 2.8)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.06d0)) .or. (.not. (x <= 2.8d0))) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.06) || !(x <= 2.8)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.06) or not (x <= 2.8):
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	else:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.06) || !(x <= 2.8))
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.06) || ~((x <= 2.8)))
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	else
		tmp = 1.6316775383 + (x * -2.134856267379707);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.06], N[Not[LessEqual[x, 2.8]], $MachinePrecision]], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0600000000000001 or 2.7999999999999998 < 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. Taylor expanded in x around inf 98.8%

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

   if -1.0600000000000001 < x < 2.7999999999999998

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. *-commutative 99.9%

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

      5. neg-mul-1 99.9%

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

      6. associate-*r* 99.9%

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

      7. *-commutative 99.9%

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

      8. fma-def 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*l/ 99.9%

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

      11. *-commutative 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-lft-in 99.9%

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

      14. *-commutative 99.9%

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

      15. associate-*r* 99.9%

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

      16. *-commutative 99.9%

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

      17. fma-def 99.9%

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

      18. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      19. metadata-eval 99.9%

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

      20. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto 1.6316775383 + \color{blue}{x \cdot -2.134856267379707} \]

   6. Simplified 99.2%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot -2.134856267379707} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \]

Alternative 6: 98.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (/ x -1.4142071247754946)
   (if (<= x 1.15)
     (+ 1.6316775383 (* x -2.134856267379707))
     (/ x -1.4142071247754946))))

C

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x / -1.4142071247754946;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = x / -1.4142071247754946;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.06d0)) then
        tmp = x / (-1.4142071247754946d0)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = x / (-1.4142071247754946d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x / -1.4142071247754946;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = x / -1.4142071247754946;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = x / -1.4142071247754946
	elif x <= 1.15:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = x / -1.4142071247754946
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.06)
		tmp = Float64(x / -1.4142071247754946);
	elseif (x <= 1.15)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(x / -1.4142071247754946);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.06)
		tmp = x / -1.4142071247754946;
	elseif (x <= 1.15)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = x / -1.4142071247754946;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.06], N[(x / -1.4142071247754946), $MachinePrecision], If[LessEqual[x, 1.15], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(x / -1.4142071247754946), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0600000000000001 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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. *-commutative 99.8%

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

      5. neg-mul-1 99.8%

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

      6. associate-*r* 99.8%

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

      7. *-commutative 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. distribute-lft-in 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-*r* 99.8%

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

      16. *-commutative 99.8%

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

      17. fma-def 99.8%

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

      18. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      19. metadata-eval 99.8%

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

      20. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. flip-+ 48.9%

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

      3. clear-num 48.8%

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

      4. swap-sqr 48.6%

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

      5. metadata-eval 48.6%

         \[\leadsto \frac{1}{\frac{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{\left(x \cdot x\right) \cdot \color{blue}{0.5000045521} - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]

      6. pow2 48.6%

         \[\leadsto \frac{1}{\frac{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{\left(x \cdot x\right) \cdot 0.5000045521 - \color{blue}{{\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}}} \]

   5. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{\left(x \cdot x\right) \cdot 0.5000045521 - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}}} \]

   6. Taylor expanded in x around inf 98.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1.4142071247754946}{x}}} \]
   7. Step-by-step derivation

      1. clear-num 98.6%

         \[\leadsto \color{blue}{\frac{x}{-1.4142071247754946}} \]

      2. frac-2neg 98.6%

         \[\leadsto \color{blue}{\frac{-x}{--1.4142071247754946}} \]

      3. neg-sub0 98.6%

         \[\leadsto \frac{\color{blue}{0 - x}}{--1.4142071247754946} \]

      4. div-sub 98.6%

         \[\leadsto \color{blue}{\frac{0}{--1.4142071247754946} - \frac{x}{--1.4142071247754946}} \]

      5. metadata-eval 98.6%

         \[\leadsto \frac{0}{\color{blue}{1.4142071247754946}} - \frac{x}{--1.4142071247754946} \]

      6. metadata-eval 98.6%

         \[\leadsto \color{blue}{0} - \frac{x}{--1.4142071247754946} \]

      7. metadata-eval 98.6%

         \[\leadsto 0 - \frac{x}{\color{blue}{1.4142071247754946}} \]

   8. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{0 - \frac{x}{1.4142071247754946}} \]
   9. Step-by-step derivation

      1. metadata-eval 98.6%

         \[\leadsto \color{blue}{\frac{0}{-1.4142071247754946}} - \frac{x}{1.4142071247754946} \]

      2. remove-double-neg 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \frac{\color{blue}{-\left(-x\right)}}{1.4142071247754946} \]

      3. mul-1-neg 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \frac{\color{blue}{-1 \cdot \left(-x\right)}}{1.4142071247754946} \]

      4. *-commutative 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \frac{\color{blue}{\left(-x\right) \cdot -1}}{1.4142071247754946} \]

      5. associate-/l* 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \color{blue}{\frac{-x}{\frac{1.4142071247754946}{-1}}} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \frac{-x}{\color{blue}{-1.4142071247754946}} \]

      7. div-sub 98.6%

         \[\leadsto \color{blue}{\frac{0 - \left(-x\right)}{-1.4142071247754946}} \]

      8. neg-sub0 98.6%

         \[\leadsto \frac{\color{blue}{-\left(-x\right)}}{-1.4142071247754946} \]

      9. remove-double-neg 98.6%

         \[\leadsto \frac{\color{blue}{x}}{-1.4142071247754946} \]

   10. Simplified 98.6%

       \[\leadsto \color{blue}{\frac{x}{-1.4142071247754946}} \]

   if -1.0600000000000001 < 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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. *-commutative 99.9%

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

      5. neg-mul-1 99.9%

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

      6. associate-*r* 99.9%

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

      7. *-commutative 99.9%

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

      8. fma-def 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*l/ 99.9%

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

      11. *-commutative 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-lft-in 99.9%

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

      14. *-commutative 99.9%

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

      15. associate-*r* 99.9%

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

      16. *-commutative 99.9%

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

      17. fma-def 99.9%

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

      18. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      19. metadata-eval 99.9%

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

      20. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto 1.6316775383 + \color{blue}{x \cdot -2.134856267379707} \]

   6. Simplified 99.2%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot -2.134856267379707} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \end{array} \]

Alternative 7: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.06) (* x -0.70711) (if (<= x 1.16) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x * -0.70711;
	} else if (x <= 1.16) {
		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 <= (-1.06d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.16d0) 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 <= -1.06) {
		tmp = x * -0.70711;
	} else if (x <= 1.16) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = x * -0.70711
	elif x <= 1.16:
		tmp = 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.06)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.16)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.06)
		tmp = x * -0.70711;
	elseif (x <= 1.16)
		tmp = 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.06], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.16], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0600000000000001 or 1.15999999999999992 < 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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. *-commutative 99.8%

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

      5. neg-mul-1 99.8%

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

      6. associate-*r* 99.8%

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

      7. *-commutative 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. distribute-lft-in 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-*r* 99.8%

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

      16. *-commutative 99.8%

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

      17. fma-def 99.8%

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

      18. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      19. metadata-eval 99.8%

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

      20. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]
   5. Step-by-step derivation

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   if -1.0600000000000001 < x < 1.15999999999999992

   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. Taylor expanded in x around 0 99.3%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} - x\right) \]
   3. Step-by-step derivation

      1. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around 0 97.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 8: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (/ x -1.4142071247754946)
   (if (<= x 1.16) 1.6316775383 (/ x -1.4142071247754946))))

C

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x / -1.4142071247754946;
	} else if (x <= 1.16) {
		tmp = 1.6316775383;
	} else {
		tmp = x / -1.4142071247754946;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.06d0)) then
        tmp = x / (-1.4142071247754946d0)
    else if (x <= 1.16d0) then
        tmp = 1.6316775383d0
    else
        tmp = x / (-1.4142071247754946d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x / -1.4142071247754946;
	} else if (x <= 1.16) {
		tmp = 1.6316775383;
	} else {
		tmp = x / -1.4142071247754946;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = x / -1.4142071247754946
	elif x <= 1.16:
		tmp = 1.6316775383
	else:
		tmp = x / -1.4142071247754946
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.06)
		tmp = Float64(x / -1.4142071247754946);
	elseif (x <= 1.16)
		tmp = 1.6316775383;
	else
		tmp = Float64(x / -1.4142071247754946);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.06)
		tmp = x / -1.4142071247754946;
	elseif (x <= 1.16)
		tmp = 1.6316775383;
	else
		tmp = x / -1.4142071247754946;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.06], N[(x / -1.4142071247754946), $MachinePrecision], If[LessEqual[x, 1.16], 1.6316775383, N[(x / -1.4142071247754946), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0600000000000001 or 1.15999999999999992 < 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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. *-commutative 99.8%

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

      5. neg-mul-1 99.8%

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

      6. associate-*r* 99.8%

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

      7. *-commutative 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

      11. *-commutative 99.8%

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

      12. +-commutative 99.8%

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

      13. distribute-lft-in 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-*r* 99.8%

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

      16. *-commutative 99.8%

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

      17. fma-def 99.8%

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

      18. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      19. metadata-eval 99.8%

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

      20. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. flip-+ 48.9%

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

      3. clear-num 48.8%

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

      4. swap-sqr 48.6%

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

      5. metadata-eval 48.6%

         \[\leadsto \frac{1}{\frac{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{\left(x \cdot x\right) \cdot \color{blue}{0.5000045521} - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]

      6. pow2 48.6%

         \[\leadsto \frac{1}{\frac{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{\left(x \cdot x\right) \cdot 0.5000045521 - \color{blue}{{\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}}} \]

   5. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{\left(x \cdot x\right) \cdot 0.5000045521 - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}}} \]

   6. Taylor expanded in x around inf 98.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1.4142071247754946}{x}}} \]
   7. Step-by-step derivation

      1. clear-num 98.6%

         \[\leadsto \color{blue}{\frac{x}{-1.4142071247754946}} \]

      2. frac-2neg 98.6%

         \[\leadsto \color{blue}{\frac{-x}{--1.4142071247754946}} \]

      3. neg-sub0 98.6%

         \[\leadsto \frac{\color{blue}{0 - x}}{--1.4142071247754946} \]

      4. div-sub 98.6%

         \[\leadsto \color{blue}{\frac{0}{--1.4142071247754946} - \frac{x}{--1.4142071247754946}} \]

      5. metadata-eval 98.6%

         \[\leadsto \frac{0}{\color{blue}{1.4142071247754946}} - \frac{x}{--1.4142071247754946} \]

      6. metadata-eval 98.6%

         \[\leadsto \color{blue}{0} - \frac{x}{--1.4142071247754946} \]

      7. metadata-eval 98.6%

         \[\leadsto 0 - \frac{x}{\color{blue}{1.4142071247754946}} \]

   8. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{0 - \frac{x}{1.4142071247754946}} \]
   9. Step-by-step derivation

      1. metadata-eval 98.6%

         \[\leadsto \color{blue}{\frac{0}{-1.4142071247754946}} - \frac{x}{1.4142071247754946} \]

      2. remove-double-neg 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \frac{\color{blue}{-\left(-x\right)}}{1.4142071247754946} \]

      3. mul-1-neg 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \frac{\color{blue}{-1 \cdot \left(-x\right)}}{1.4142071247754946} \]

      4. *-commutative 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \frac{\color{blue}{\left(-x\right) \cdot -1}}{1.4142071247754946} \]

      5. associate-/l* 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \color{blue}{\frac{-x}{\frac{1.4142071247754946}{-1}}} \]

      6. metadata-eval 98.6%

         \[\leadsto \frac{0}{-1.4142071247754946} - \frac{-x}{\color{blue}{-1.4142071247754946}} \]

      7. div-sub 98.6%

         \[\leadsto \color{blue}{\frac{0 - \left(-x\right)}{-1.4142071247754946}} \]

      8. neg-sub0 98.6%

         \[\leadsto \frac{\color{blue}{-\left(-x\right)}}{-1.4142071247754946} \]

      9. remove-double-neg 98.6%

         \[\leadsto \frac{\color{blue}{x}}{-1.4142071247754946} \]

   10. Simplified 98.6%

       \[\leadsto \color{blue}{\frac{x}{-1.4142071247754946}} \]

   if -1.0600000000000001 < x < 1.15999999999999992

   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. Taylor expanded in x around 0 99.3%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} - x\right) \]
   3. Step-by-step derivation

      1. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around 0 97.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \end{array} \]

Alternative 9: 51.2% accurate, 19.0× 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. Taylor expanded in x around 0 98.8%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} - x\right) \]
3. Step-by-step derivation

   1. *-commutative 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around 0 59.1%

   \[\leadsto \color{blue}{1.6316775383} \]

6. Final simplification 59.1%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023297 
(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)))