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

Average Error: 0.1 → 0.1

Time: 8.2s

Precision: binary64

Cost: 26304

Math

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

↓

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

FPCore

(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
  -0.70711
  (/ (fma x 0.1913510371 1.6316775383) (fma x (fma x 0.04481 0.99229) 1.0))))

C

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, -0.70711, (fma(x, 0.1913510371, 1.6316775383) / fma(x, fma(x, 0.04481, 0.99229), 1.0)));
}

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

↓

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

Wolfram

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

Derivation

1. Initial program 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. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   Proof
   (fma.f64 x -70711/100000 (/.f64 (fma.f64 x 1913510371/10000000000 16316775383/10000000000) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (Rewrite<= metadata-eval (*.f64 70711/100000 -1)) (/.f64 (fma.f64 x 1913510371/10000000000 16316775383/10000000000) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (fma.f64 x (Rewrite<= metadata-eval (*.f64 27061/100000 70711/100000)) 16316775383/10000000000) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (fma.f64 x (*.f64 27061/100000 70711/100000) (Rewrite<= metadata-eval (*.f64 230753/100000 70711/100000))) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (*.f64 27061/100000 70711/100000)) (*.f64 230753/100000 70711/100000))) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 27061/100000) 70711/100000)) (*.f64 230753/100000 70711/100000)) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 230753/100000 70711/100000) (*.f64 (*.f64 x 27061/100000) 70711/100000))) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 70711/100000 (+.f64 230753/100000 (*.f64 x 27061/100000)))) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 2 points increase in error, 1 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (*.f64 70711/100000 (+.f64 230753/100000 (*.f64 x 27061/100000))) (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 4481/100000) 99229/100000)) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (*.f64 70711/100000 (+.f64 230753/100000 (*.f64 x 27061/100000))) (fma.f64 x (Rewrite<= +-commutative_binary64 (+.f64 99229/100000 (*.f64 x 4481/100000))) 1))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (*.f64 70711/100000 (+.f64 230753/100000 (*.f64 x 27061/100000))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))) 1)))): 0 points increase in error, 1 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (*.f64 70711/100000 (+.f64 230753/100000 (*.f64 x 27061/100000))) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (Rewrite<= associate-*r/_binary64 (*.f64 70711/100000 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000)))))))): 3 points increase in error, 1 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (*.f64 70711/100000 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000)))))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (*.f64 70711/100000 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (neg.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000)))))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 70711/100000 -1) (neg.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000)))))) (*.f64 70711/100000 -1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (*.f64 70711/100000 -1)) (*.f64 (neg.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000)))))) (*.f64 70711/100000 -1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-in_binary64 (*.f64 (*.f64 70711/100000 -1) (+.f64 x (neg.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))))))): 2 points increase in error, 2 points decrease in error
   (*.f64 (*.f64 70711/100000 -1) (Rewrite<= sub-neg_binary64 (-.f64 x (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000)))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r*_binary64 (*.f64 70711/100000 (*.f64 -1 (-.f64 x (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 70711/100000 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 x (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 70711/100000 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 x (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 70711/100000 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 x) (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 70711/100000 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 x)) (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 70711/100000 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))) (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (*.f64 70711/100000 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))) x))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	7488

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

Alternative 2
Error	0.5
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;\frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 + \frac{-58.14938538768042}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;\left(1.6316775383 + x \cdot -2.134856267379707\right) + \left(x \cdot x\right) \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]

Alternative 3
Error	0.1
Cost	1216

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

Alternative 4
Error	0.6
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\left(1.6316775383 + x \cdot -2.134856267379707\right) + \left(x \cdot x\right) \cdot 1.3436228731669864\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]

Alternative 5
Error	0.6
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 + \frac{-58.14938538768042}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\left(1.6316775383 + x \cdot -2.134856267379707\right) + \left(x \cdot x\right) \cdot 1.3436228731669864\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]

Alternative 6
Error	0.8
Cost	712

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

Alternative 7
Error	0.7
Cost	712

\[\begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	0.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.55:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]

Alternative 9
Error	0.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 10
Error	26.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;x \cdot -2.134856267379707\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707\\ \end{array} \]

Alternative 11
Error	1.2
Cost	456

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

Alternative 12
Error	31.2
Cost	64

\[1.6316775383 \]

Reproduce

herbie shell --seed 2022329 
(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)))