Jmat.Real.erfi, branch x greater than or equal to 5

Average Error: 2.8 → 1.2

Time: 16.7s

Precision: binary64

Cost: 46272

\[x \geq 0.5\]

Math

\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

↓

\[\frac{1}{\sqrt{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right)\right)\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
  (+
   (+
    (+
     (/ 1.0 (fabs x))
     (*
      (/ 1.0 2.0)
      (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))
    (*
     (/ 3.0 4.0)
     (*
      (*
       (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))
       (/ 1.0 (fabs x)))
      (/ 1.0 (fabs x)))))
   (*
    (/ 15.0 8.0)
    (*
     (*
      (*
       (*
        (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))
        (/ 1.0 (fabs x)))
       (/ 1.0 (fabs x)))
      (/ 1.0 (fabs x)))
     (/ 1.0 (fabs x)))))))

↓

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (*
   (/ (pow (exp x) x) x)
   (+ 1.0 (fma 1.875 (pow x -6.0) (+ (/ 0.75 (pow x 4.0)) (/ 0.5 (* x x))))))))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * ((((1.0 / fabs(x)) + ((1.0 / 2.0) * (((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))))) + ((3.0 / 4.0) * (((((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))))) + ((15.0 / 8.0) * (((((((1.0 / fabs(x)) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x))) * (1.0 / fabs(x)))));
}

↓

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * ((pow(exp(x), x) / x) * (1.0 + fma(1.875, pow(x, -6.0), ((0.75 / pow(x, 4.0)) + (0.5 / (x * x))))));
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(Float64(1.0 / abs(x)) + Float64(Float64(1.0 / 2.0) * Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))) + Float64(Float64(3.0 / 4.0) * Float64(Float64(Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))) + Float64(Float64(15.0 / 8.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 / abs(x)) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))) * Float64(1.0 / abs(x))))))
end

↓

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64((exp(x) ^ x) / x) * Float64(1.0 + fma(1.875, (x ^ -6.0), Float64(Float64(0.75 / (x ^ 4.0)) + Float64(0.5 / Float64(x * x)))))))
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / x), $MachinePrecision] * N[(1.0 + N[(1.875 * N[Power[x, -6.0], $MachinePrecision] + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.8

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Simplified 2.7

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]
   Proof

   [Start] 2.8

   \[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   associate-+l+ [=>] 2.8

   \[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Applied egg-rr 1.3

   \[\leadsto \color{blue}{\frac{\frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(0.5 + 0.75 \cdot {x}^{-2}\right) \cdot {x}^{-2}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}}} \]

4. Simplified 1.2

   \[\leadsto \color{blue}{\frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, {x}^{-2} \cdot \left(0.5 + 0.75 \cdot {x}^{-2}\right)\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}}} \]
   Proof

   [Start] 1.3	\[ \frac{\frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(0.5 + 0.75 \cdot {x}^{-2}\right) \cdot {x}^{-2}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-/l/ [=>] 1.2	\[ \color{blue}{\frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(0.5 + 0.75 \cdot {x}^{-2}\right) \cdot {x}^{-2}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}}} \]
   fma-def [=>] 1.2	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \color{blue}{\mathsf{fma}\left(1.875, {x}^{-6}, \left(0.5 + 0.75 \cdot {x}^{-2}\right) \cdot {x}^{-2}\right)}\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]
   *-commutative [=>] 1.2	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \color{blue}{{x}^{-2} \cdot \left(0.5 + 0.75 \cdot {x}^{-2}\right)}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]

5. Taylor expanded in x around 0 1.2

   \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \color{blue}{0.75 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]

6. Simplified 1.2

   \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \color{blue}{\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]
   Proof

   [Start] 1.2	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, 0.75 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]
   associate-*r/ [=>] 1.2	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \color{blue}{\frac{0.75 \cdot 1}{{x}^{4}}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]
   metadata-eval [=>] 1.2	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \frac{\color{blue}{0.75}}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]
   associate-*r/ [=>] 1.2	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \frac{0.75}{{x}^{4}} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]
   metadata-eval [=>] 1.2	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \frac{0.75}{{x}^{4}} + \frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]
   unpow2 [=>] 1.2	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \frac{0.75}{{x}^{4}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]

7. Applied egg-rr 1.2

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right)\right)\right)} \]

8. Final simplification 1.2

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right)\right)\right) \]

Alternatives

Alternative 1
Error	1.2
Cost	40128

\[\sqrt{\frac{1}{\pi}} \cdot \left({\left(e^{x}\right)}^{x} \cdot \left(\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right) + \frac{\frac{0.5}{x} + \frac{1.875}{{x}^{5}}}{x \cdot x}\right)\right) \]

Alternative 2
Error	1.2
Cost	33664

\[\frac{\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}}{x} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

Alternative 3
Error	1.3
Cost	33536

\[\frac{\frac{{\left(e^{x}\right)}^{x}}{x}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right) \]

Alternative 4
Error	41.5
Cost	33216

\[e^{x \cdot x} \cdot \frac{\frac{0.75}{{x}^{5}} + \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \]

Alternative 5
Error	44.7
Cost	33088

\[\begin{array}{l} t_0 := e^{x \cdot x}\\ \frac{1}{\sqrt{\pi}} \cdot \left(\frac{t_0}{x} + 0.5 \cdot \frac{t_0}{{x}^{3}}\right) \end{array} \]

Alternative 6
Error	48.0
Cost	32896

\[\sqrt{\frac{1}{\pi}} \cdot \left({\left(e^{x}\right)}^{x} \cdot \left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right)\right) \]

Alternative 7
Error	48.0
Cost	32896

\[\left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right) \cdot \left({\left(e^{x}\right)}^{x} \cdot \sqrt{\frac{1}{\pi}}\right) \]

Alternative 8
Error	48.3
Cost	26048

\[\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \sqrt{\frac{1}{\pi}} \]

Alternative 9
Error	48.3
Cost	19712

\[\sqrt{\frac{1}{\pi}} \cdot \frac{e^{x \cdot x}}{x} \]

Alternative 10
Error	56.7
Cost	13184

\[\sqrt{\frac{1}{\pi}} \cdot \frac{1.5}{x} \]

Reproduce

herbie shell --seed 2023053 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))