Average Error: 0.1 → 0.1
Time: 8.7s
Precision: binary64
Cost: 13888
\[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
↓
\[\mathsf{fma}\left(\frac{x \cdot 0.27061 + 2.30753}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)}, 0.70711, x \cdot -0.70711\right) \]
(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))
↓
(FPCore (x)
 :precision binary64
 (fma
  (/ (+ (* x 0.27061) 2.30753) (fma x (+ (* x 0.04481) 0.99229) 1.0))
  0.70711
  (* x -0.70711)))
double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}
↓
double code(double x) {
	return fma((((x * 0.27061) + 2.30753) / fma(x, ((x * 0.04481) + 0.99229), 1.0)), 0.70711, (x * -0.70711));
}

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

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

0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right)
↓
\mathsf{fma}\left(\frac{x \cdot 0.27061 + 2.30753}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)}, 0.70711, x \cdot -0.70711\right)

Error

Derivation

  1. Initial program 0.1

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

  2. Applied egg-rr0.1

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

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

  5. Taylor expanded in x around 0 0.1

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

  6. Simplified0.1

    \[\leadsto \mathsf{fma}\left(\frac{x \cdot 0.27061 + 2.30753}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)}, 0.70711, \color{blue}{x \cdot -0.70711}\right) \]
    Proof
    (*.f64 x -70711/100000): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-commutative_binary64 (*.f64 -70711/100000 x)): 0 points increase in error, 0 points decrease in error

  7. Final simplification0.1

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

Alternatives

Alternative 1
Error0.1
Cost13760
\[\mathsf{fma}\left(x, -0.70711, \frac{x \cdot 0.1913510371 + 1.6316775383}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)}\right) \]
Alternative 2
Error0.1
Cost1216
\[0.70711 \cdot \left(\frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot \left(x \cdot 0.04481 + 0.99229\right)} - x\right) \]
Alternative 3
Error0.8
Cost964
\[\begin{array}{l} \mathbf{if}\;x \leq -27.95234177957094:\\ \;\;\;\;0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} + \frac{-82.23527511657367}{x \cdot x}\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0014684735685288471:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
Alternative 4
Error0.9
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -27.95234177957094:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.0014684735685288471:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
Alternative 5
Error0.9
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -27.95234177957094:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 0.0014684735685288471:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
Alternative 6
Error1.3
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -27.95234177957094:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.0014684735685288471:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
Alternative 7
Error32.0
Cost64
\[1.6316775383 \]

Error

Reproduce

herbie shell --seed 2022316 
(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)))