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

Average Error: 2.8 → 1.3

Time: 5.8s

Precision: binary64

\[x \geq 0.5\]

Math

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

↓

\[\sqrt{\frac{1}{\pi}} \cdot \left(\frac{e^{{x}^{2}}}{{x}^{5}} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right) + \frac{\frac{\left(0.015625 \cdot {\left({x}^{-4}\right)}^{3} + -1\right) \cdot {\left(e^{x}\right)}^{x}}{\frac{0.0625}{{x}^{8}} + \mathsf{fma}\left(0.25, {x}^{-4}, 1\right)}}{x \cdot \mathsf{fma}\left(0.5, {x}^{-2}, -1\right)}\right) \]

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
 (*
  (sqrt (/ 1.0 PI))
  (+
   (* (/ (exp (pow x 2.0)) (pow x 5.0)) (+ 0.75 (/ 1.875 (* x x))))
   (/
    (/
     (* (+ (* 0.015625 (pow (pow x -4.0) 3.0)) -1.0) (pow (exp x) x))
     (+ (/ 0.0625 (pow x 8.0)) (fma 0.25 (pow x -4.0) 1.0)))
    (* x (fma 0.5 (pow x -2.0) -1.0))))))

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 sqrt((1.0 / ((double) M_PI))) * (((exp(pow(x, 2.0)) / pow(x, 5.0)) * (0.75 + (1.875 / (x * x)))) + (((((0.015625 * pow(pow(x, -4.0), 3.0)) + -1.0) * pow(exp(x), x)) / ((0.0625 / pow(x, 8.0)) + fma(0.25, pow(x, -4.0), 1.0))) / (x * fma(0.5, pow(x, -2.0), -1.0))));
}

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(sqrt(Float64(1.0 / pi)) * Float64(Float64(Float64(exp((x ^ 2.0)) / (x ^ 5.0)) * Float64(0.75 + Float64(1.875 / Float64(x * x)))) + Float64(Float64(Float64(Float64(Float64(0.015625 * ((x ^ -4.0) ^ 3.0)) + -1.0) * (exp(x) ^ x)) / Float64(Float64(0.0625 / (x ^ 8.0)) + fma(0.25, (x ^ -4.0), 1.0))) / Float64(x * fma(0.5, (x ^ -2.0), -1.0)))))
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[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.015625 * N[Power[N[Power[x, -4.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] / N[(N[(0.0625 / N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[Power[x, -4.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(0.5 * N[Power[x, -2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

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 1.4

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

3. Taylor expanded in x around inf 2.7

   \[\leadsto \color{blue}{0.75 \cdot \left(\frac{e^{{x}^{2}}}{{\left(\left|x\right|\right)}^{5}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(1.875 \cdot \left(\frac{e^{{x}^{2}}}{{\left(\left|x\right|\right)}^{5} \cdot {x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(\frac{e^{{x}^{2}}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}} + 0.5 \cdot \left(\frac{e^{{x}^{2}}}{\left|x\right| \cdot {x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]

4. Simplified 1.3

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

5. Applied egg-rr 1.3

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{{x}^{5}} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right) + \color{blue}{\frac{\left(\frac{0.25}{{x}^{4}} - 1\right) \cdot {\left(e^{x}\right)}^{x}}{\mathsf{fma}\left(0.5, {x}^{-2}, -1\right) \cdot x}}\right) \]

6. Applied egg-rr 1.3

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{{x}^{5}} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right) + \frac{\color{blue}{\frac{\left(0.015625 \cdot {\left({x}^{-4}\right)}^{3} + -1\right) \cdot {\left(e^{x}\right)}^{x}}{\frac{0.0625}{{x}^{8}} + \mathsf{fma}\left(0.25, {x}^{-4}, 1\right)}}}{\mathsf{fma}\left(0.5, {x}^{-2}, -1\right) \cdot x}\right) \]

7. Taylor expanded in x around inf 1.3

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{e^{{x}^{2}}}}{{x}^{5}} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right) + \frac{\frac{\left(0.015625 \cdot {\left({x}^{-4}\right)}^{3} + -1\right) \cdot {\left(e^{x}\right)}^{x}}{\frac{0.0625}{{x}^{8}} + \mathsf{fma}\left(0.25, {x}^{-4}, 1\right)}}{\mathsf{fma}\left(0.5, {x}^{-2}, -1\right) \cdot x}\right) \]

8. Final simplification 1.3

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{e^{{x}^{2}}}{{x}^{5}} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right) + \frac{\frac{\left(0.015625 \cdot {\left({x}^{-4}\right)}^{3} + -1\right) \cdot {\left(e^{x}\right)}^{x}}{\frac{0.0625}{{x}^{8}} + \mathsf{fma}\left(0.25, {x}^{-4}, 1\right)}}{x \cdot \mathsf{fma}\left(0.5, {x}^{-2}, -1\right)}\right) \]

Reproduce

herbie shell --seed 2022153 
(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)))))))