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

Average Error: 0.1 → 0.1

Time: 9.9s

Precision: binary64

Cost: 14144

Math

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

↓

\[-1 + \left(x \cdot -0.70711 + e^{\mathsf{log1p}\left(\frac{-1.6316775383 + x \cdot -0.1913510371}{-1 + x \cdot \left(x \cdot -0.04481 + -0.99229\right)}\right)}\right) \]

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
 (+
  -1.0
  (+
   (* x -0.70711)
   (exp
    (log1p
     (/
      (+ -1.6316775383 (* x -0.1913510371))
      (+ -1.0 (* x (+ (* x -0.04481) -0.99229)))))))))

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 -1.0 + ((x * -0.70711) + exp(log1p(((-1.6316775383 + (x * -0.1913510371)) / (-1.0 + (x * ((x * -0.04481) + -0.99229)))))));
}

Java

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

↓

public static double code(double x) {
	return -1.0 + ((x * -0.70711) + Math.exp(Math.log1p(((-1.6316775383 + (x * -0.1913510371)) / (-1.0 + (x * ((x * -0.04481) + -0.99229)))))));
}

Python

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

↓

def code(x):
	return -1.0 + ((x * -0.70711) + math.exp(math.log1p(((-1.6316775383 + (x * -0.1913510371)) / (-1.0 + (x * ((x * -0.04481) + -0.99229)))))))

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 Float64(-1.0 + Float64(Float64(x * -0.70711) + exp(log1p(Float64(Float64(-1.6316775383 + Float64(x * -0.1913510371)) / Float64(-1.0 + Float64(x * Float64(Float64(x * -0.04481) + -0.99229))))))))
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[(-1.0 + N[(N[(x * -0.70711), $MachinePrecision] + N[Exp[N[Log[1 + N[(N[(-1.6316775383 + N[(x * -0.1913510371), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(x * N[(N[(x * -0.04481), $MachinePrecision] + -0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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 70711/100000 27061/100000)) 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 70711/100000 27061/100000) (Rewrite<= metadata-eval (*.f64 70711/100000 230753/100000))) (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 70711/100000 27061/100000)) (*.f64 70711/100000 230753/100000))) (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=> *-commutative_binary64 (*.f64 (*.f64 70711/100000 27061/100000) x)) (*.f64 70711/100000 230753/100000)) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 24 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 70711/100000 (*.f64 27061/100000 x))) (*.f64 70711/100000 230753/100000)) (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 (*.f64 70711/100000 (Rewrite<= *-commutative_binary64 (*.f64 x 27061/100000))) (*.f64 70711/100000 230753/100000)) (fma.f64 x (fma.f64 x 4481/100000 99229/100000) 1))): 0 points increase in error, 24 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 70711/100000 (+.f64 (*.f64 x 27061/100000) 230753/100000))) (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 70711/100000 (Rewrite<= +-commutative_binary64 (+.f64 230753/100000 (*.f64 x 27061/100000)))) (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 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, 0 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)))))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x (*.f64 70711/100000 -1) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))) 70711/100000))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (*.f64 70711/100000 -1)) (*.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))) 70711/100000))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 70711/100000 -1) x)) (*.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))) 70711/100000)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 70711/100000 (*.f64 -1 x))) (*.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))) 70711/100000)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 70711/100000 (Rewrite<= neg-mul-1_binary64 (neg.f64 x))) (*.f64 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000))))) 70711/100000)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 70711/100000 (neg.f64 x)) (Rewrite=> *-commutative_binary64 (*.f64 70711/100000 (/.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<= +-commutative_binary64 (+.f64 (*.f64 70711/100000 (/.f64 (+.f64 230753/100000 (*.f64 x 27061/100000)) (+.f64 1 (*.f64 x (+.f64 99229/100000 (*.f64 x 4481/100000)))))) (*.f64 70711/100000 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-in_binary64 (*.f64 70711/100000 (+.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. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\left(x \cdot -0.70711 + e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}\right) - 1} \]

4. Applied egg-rr 0.1

   \[\leadsto \left(x \cdot -0.70711 + e^{\mathsf{log1p}\left(\color{blue}{-\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}}\right)}\right) - 1 \]

5. Simplified 0.1

   \[\leadsto \left(x \cdot -0.70711 + e^{\mathsf{log1p}\left(\color{blue}{\frac{-1.6316775383 + x \cdot -0.1913510371}{-1 - x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right)}}\right)}\right) - 1 \]
   Proof
   (-.f64 (+.f64 (*.f64 x -70711/100000) (exp.f64 (log1p.f64 (/.f64 (+.f64 -16316775383/10000000000 (*.f64 x -1913510371/10000000000)) (-.f64 -1 (*.f64 x (fma.f64 x 4481/100000 99229/100000))))))) 1): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 x -70711/100000) (exp.f64 (log1p.f64 (/.f64 (+.f64 (Rewrite<= metadata-eval (neg.f64 16316775383/10000000000)) (*.f64 x -1913510371/10000000000)) (-.f64 -1 (*.f64 x (fma.f64 x 4481/100000 99229/100000))))))) 1): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 x -70711/100000) (exp.f64 (log1p.f64 (/.f64 (+.f64 (neg.f64 16316775383/10000000000) (*.f64 x (Rewrite<= metadata-eval (neg.f64 1913510371/10000000000)))) (-.f64 -1 (*.f64 x (fma.f64 x 4481/100000 99229/100000))))))) 1): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 x -70711/100000) (exp.f64 (log1p.f64 (/.f64 (+.f64 (neg.f64 16316775383/10000000000) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x 1913510371/10000000000)))) (-.f64 -1 (*.f64 x (fma.f64 x 4481/100000 99229/100000))))))) 1): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 x -70711/100000) (exp.f64 (log1p.f64 (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 16316775383/10000000000 (*.f64 x 1913510371/10000000000)))) (-.f64 -1 (*.f64 x (fma.f64 x 4481/100000 99229/100000))))))) 1): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 x -70711/100000) (exp.f64 (log1p.f64 (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x 1913510371/10000000000) 16316775383/10000000000))) (-.f64 -1 (*.f64 x (fma.f64 x 4481/100000 99229/100000))))))) 1): 8 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 x -70711/100000) (exp.f64 (log1p.f64 (/.f64 (neg.f64 (Rewrite=> fma-def_binary64 (fma.f64 x 1913510371/10000000000 16316775383/10000000000))) (-.f64 -1 (*.f64 x (fma.f64 x 4481/100000 99229/100000))))))) 1): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (*.f64 x -70711/100000) (exp.f64 (log1p.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (fma.f64 x 1913510371/10000000000 16316775383/10000000000) (-.f64 -1 (*.f64 x (fma.f64 x 4481/100000 99229/100000))))))))) 1): 0 points increase in error, 8 points decrease in error

6. Applied egg-rr 0.1

   \[\leadsto \left(x \cdot -0.70711 + e^{\mathsf{log1p}\left(\frac{-1.6316775383 + x \cdot -0.1913510371}{-1 - x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)}}\right)}\right) - 1 \]

7. Final simplification 0.1

   \[\leadsto -1 + \left(x \cdot -0.70711 + e^{\mathsf{log1p}\left(\frac{-1.6316775383 + x \cdot -0.1913510371}{-1 + x \cdot \left(x \cdot -0.04481 + -0.99229\right)}\right)}\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	7488

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

Alternative 2
Error	0.1
Cost	1600

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

Alternative 3
Error	0.1
Cost	1216

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

Alternative 4
Error	0.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} - \frac{\frac{82.23527511657367}{x}}{x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(x \cdot \left(x \cdot 2.003561459544073 + -3.0191289437\right)\right) + 1.6316775383\\ \end{array} \]

Alternative 5
Error	0.6
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} - \frac{\frac{82.23527511657367}{x}}{x}\right) - x\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;0.70711 \cdot \left(x \cdot \left(x \cdot 2.003561459544073 + -3.0191289437\right)\right) + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 + \frac{-58.14938538768042}{x \cdot x}\right)\\ \end{array} \]

Alternative 6
Error	0.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 0.74\right):\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(x \cdot -2.0191289437\right) + 0.70711 \cdot \left(2.30753 - x\right)\\ \end{array} \]

Alternative 7
Error	0.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(x \cdot \left(x \cdot 2.003561459544073 + -3.0191289437\right)\right) + 1.6316775383\\ \end{array} \]

Alternative 8
Error	1.0
Cost	960

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

Alternative 9
Error	0.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 0.74\right):\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \]

Alternative 10
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 11
Error	1.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 12
Error	31.4
Cost	64

\[1.6316775383 \]

Reproduce

herbie shell --seed 2022343 
(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)))