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

Average Error: 2.8 → 1.3

Time: 17.1s

Precision: binary64

Cost: 65472

\[x \geq 0.5\]

Math

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

↓

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

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
 (/
  (/
   (/
    (+
     1.0
     (fma 1.875 (pow x -6.0) (* (pow x -2.0) (+ 0.5 (* (pow x -2.0) 0.75)))))
    (/ x (pow (exp x) 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 (((1.0 + fma(1.875, pow(x, -6.0), (pow(x, -2.0) * (0.5 + (pow(x, -2.0) * 0.75))))) / (x / pow(exp(x), 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(Float64(1.0 + fma(1.875, (x ^ -6.0), Float64((x ^ -2.0) * Float64(0.5 + Float64((x ^ -2.0) * 0.75))))) / Float64(x / (exp(x) ^ x))) / (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[(1.0 + N[(1.875 * N[Power[x, -6.0], $MachinePrecision] + N[(N[Power[x, -2.0], $MachinePrecision] * N[(0.5 + N[(N[Power[x, -2.0], $MachinePrecision] * 0.75), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x / N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 0.25], $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. Applied egg-rr 1.3

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

5. Final simplification 1.3

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

Alternatives

Alternative 1
Error	1.2
Cost	52608

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

Alternative 2
Error	1.2
Cost	46272

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

Alternative 3
Error	1.3
Cost	46208

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

Alternative 4
Error	1.3
Cost	46208

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

Alternative 5
Error	1.3
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 6
Error	1.3
Cost	39936

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

Alternative 7
Error	1.3
Cost	39936

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

Alternative 8
Error	40.4
Cost	39748

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.12:\\ \;\;\;\;\frac{1}{\frac{x \cdot \sqrt{\pi}}{2.1875 + \left(\frac{1.875}{{x}^{6}} + \left(\frac{2.625}{{x}^{4}} + \frac{2.1875}{x \cdot x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x} \cdot \frac{\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}}{\sqrt{\pi}}\\ \end{array} \]

Alternative 9
Error	2.7
Cost	33856

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

Alternative 10
Error	2.7
Cost	33600

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

Alternative 11
Error	43.6
Cost	33540

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2:\\ \;\;\;\;\frac{1}{\frac{x \cdot \sqrt{\pi}}{2.1875 + \left(\frac{1.875}{{x}^{6}} + \left(\frac{2.625}{{x}^{4}} + \frac{2.1875}{x \cdot x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right)\\ \end{array} \]

Alternative 12
Error	44.7
Cost	26560

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

Alternative 13
Error	44.7
Cost	20224

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

Alternative 14
Error	48.3
Cost	19712

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

Alternative 15
Error	48.3
Cost	19712

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

Alternative 16
Error	56.8
Cost	19648

\[1.875 \cdot \frac{\sqrt{\frac{1}{\pi}}}{{x}^{7}} \]

Alternative 17
Error	56.9
Cost	12992

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

Reproduce

herbie shell --seed 2023073 
(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)))))))