Jmat.Real.erfi, branch x less than or equal to 0.5

Average Error: 0.2 → 0.1

Time: 8.1s

Precision: binary64

Cost: 52480

\[x \leq 0.5\]

Math

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

↓

\[\left|{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + \mathsf{fma}\left(0.2, x \cdot {x}^{4}, {x}^{6} \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+
     (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x))))
     (*
      (/ 1.0 5.0)
      (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x))))
    (*
     (/ 1.0 21.0)
     (*
      (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))
      (fabs x)))))))

↓

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (pow PI -0.5)
   (+
    (fma 2.0 x (* 0.6666666666666666 (pow x 3.0)))
    (fma 0.2 (* x (pow x 4.0)) (* (pow x 6.0) (* x 0.047619047619047616)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * ((fabs(x) * fabs(x)) * fabs(x)))) + ((1.0 / 5.0) * ((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)))) + ((1.0 / 21.0) * ((((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x))))));
}

↓

double code(double x) {
	return fabs((pow(((double) M_PI), -0.5) * (fma(2.0, x, (0.6666666666666666 * pow(x, 3.0))) + fma(0.2, (x * pow(x, 4.0)), (pow(x, 6.0) * (x * 0.047619047619047616))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * Float64(Float64(abs(x) * abs(x)) * abs(x)))) + Float64(Float64(1.0 / 5.0) * Float64(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)))) + Float64(Float64(1.0 / 21.0) * Float64(Float64(Float64(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * abs(x))))))
end

↓

function code(x)
	return abs(Float64((pi ^ -0.5) * Float64(fma(2.0, x, Float64(0.6666666666666666 * (x ^ 3.0))) + fma(0.2, Float64(x * (x ^ 4.0)), Float64((x ^ 6.0) * Float64(x * 0.047619047619047616))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(2.0 * x + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[(x * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 6.0], $MachinePrecision] * N[(x * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.2

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

2. Simplified 0.2

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right|} \]
   Proof

   [Start] 0.2

   \[ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   *-lft-identity [<=] 0.2

   \[ \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

3. Applied egg-rr 0.1

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + {\pi}^{-0.5} \cdot \mathsf{fma}\left(0.2, x \cdot {x}^{4}, {x}^{6} \cdot \left(x \cdot 0.047619047619047616\right)\right)}\right| \]

4. Simplified 0.1

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + \mathsf{fma}\left(0.2, x \cdot {x}^{4}, {x}^{6} \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)}\right| \]
   Proof

   [Start] 0.1	\[ \left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + {\pi}^{-0.5} \cdot \mathsf{fma}\left(0.2, x \cdot {x}^{4}, {x}^{6} \cdot \left(x \cdot 0.047619047619047616\right)\right)\right| \]
   distribute-lft-out [=>] 0.1	\[ \left|\color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + \mathsf{fma}\left(0.2, x \cdot {x}^{4}, {x}^{6} \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)}\right| \]

5. Final simplification 0.1

   \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + \mathsf{fma}\left(0.2, x \cdot {x}^{4}, {x}^{6} \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right| \]

Alternatives

Alternative 1
Error	0.2
Cost	45952

\[\left|{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right) + \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]

Alternative 2
Error	0.5
Cost	33216

\[\left|\frac{x}{\sqrt{\pi}} \cdot \left({x}^{4} \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right) + \mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right)\right)\right| \]

Alternative 3
Error	0.7
Cost	32836

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{6}}{\frac{\sqrt{\pi}}{x \cdot 0.047619047619047616}}\right|\\ \end{array} \]

Alternative 4
Error	1.1
Cost	32832

\[\left|\frac{x}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + {x}^{6} \cdot 0.047619047619047616\right)\right| \]

Alternative 5
Error	0.7
Cost	32580

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{6}}{\sqrt{\pi} \cdot \frac{21}{x}}\right|\\ \end{array} \]

Alternative 6
Error	0.7
Cost	32452

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 7
Error	0.7
Cost	26180

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\frac{{x}^{6}}{\frac{\sqrt{\pi}}{x \cdot 0.047619047619047616}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \]

Alternative 8
Error	2.8
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\sqrt{{x}^{14} \cdot \frac{0.0022675736961451248}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \]

Alternative 9
Error	4.4
Cost	19968

\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right| \]

Alternative 10
Error	4.7
Cost	19456

\[\left|x \cdot \frac{2}{\sqrt{\pi}}\right| \]

Reproduce

herbie shell --seed 2023017 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))