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

Average Error: 4.35% → 1.88%

Time: 16.8s

Precision: binary64

Cost: 71808

\[x \geq 0.5\]

Math

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

↓

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

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
 (/
  (/
   (*
    (pow (exp x) x)
    (-
     -1.0
     (fma 1.875 (pow x -6.0) (* (pow x -2.0) (fma (pow x -2.0) 0.75 0.5)))))
   (- x))
  (* (pow PI 0.25) (pow PI 0.25))))

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 ((pow(exp(x), x) * (-1.0 - fma(1.875, pow(x, -6.0), (pow(x, -2.0) * fma(pow(x, -2.0), 0.75, 0.5))))) / -x) / (pow(((double) M_PI), 0.25) * pow(((double) M_PI), 0.25));
}

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(Float64((exp(x) ^ x) * Float64(-1.0 - fma(1.875, (x ^ -6.0), Float64((x ^ -2.0) * fma((x ^ -2.0), 0.75, 0.5))))) / Float64(-x)) / Float64((pi ^ 0.25) * (pi ^ 0.25)))
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[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(-1.0 - N[(1.875 * N[Power[x, -6.0], $MachinePrecision] + N[(N[Power[x, -2.0], $MachinePrecision] * N[(N[Power[x, -2.0], $MachinePrecision] * 0.75 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] / N[(N[Power[Pi, 0.25], $MachinePrecision] * N[Power[Pi, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 4.35

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

2. Simplified 4.16

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

   [Start] 4.35

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

   associate-+l+ [=>] 4.35

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

3. Applied egg-rr 1.94

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

4. Simplified 1.89

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

   [Start] 1.94	\[ \frac{\frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(0.5 + 0.75 \cdot {x}^{-2}\right) \cdot {x}^{-2}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-/l/ [=>] 1.9	\[ \color{blue}{\frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(0.5 + 0.75 \cdot {x}^{-2}\right) \cdot {x}^{-2}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}}} \]
   fma-def [=>] 1.89	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \color{blue}{\mathsf{fma}\left(1.875, {x}^{-6}, \left(0.5 + 0.75 \cdot {x}^{-2}\right) \cdot {x}^{-2}\right)}\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]
   *-commutative [=>] 1.89	\[ \frac{\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \mathsf{fma}\left(1.875, {x}^{-6}, \color{blue}{{x}^{-2} \cdot \left(0.5 + 0.75 \cdot {x}^{-2}\right)}\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}} \]

5. Applied egg-rr 1.88

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

6. Final simplification 1.88

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

Alternatives

Alternative 1
Error	1.89%
Cost	65472

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

Alternative 2
Error	1.91%
Cost	46464

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

Alternative 3
Error	1.95%
Cost	39936

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

Alternative 4
Error	1.89%
Cost	33728

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

Alternative 5
Error	1.95%
Cost	33600

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

Alternative 6
Error	1.96%
Cost	33536

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

Alternative 7
Error	68.39%
Cost	32900

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.96:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(\frac{2.625}{{x}^{5}} + \frac{1.875}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi} \cdot \left(x + \frac{-0.5}{x}\right)}\\ \end{array} \]

Alternative 8
Error	68.69%
Cost	32708

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.71:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \frac{1.875}{{x}^{7}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi} \cdot \left(x + \frac{-0.5}{x}\right)}\\ \end{array} \]

Alternative 9
Error	4.16%
Cost	27200

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

Alternative 10
Error	75.38%
Cost	19712

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

Alternative 11
Error	88.34%
Cost	13184

\[\frac{x + \frac{1}{x}}{\sqrt{\pi}} \]

Alternative 12
Error	88.8%
Cost	12992

\[\frac{{\pi}^{-0.5}}{x} \]

Alternative 13
Error	97.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023103 
(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)))))))