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

Average Error: 0.23% → 0.21%

Time: 10.2s

Precision: binary64

Cost: 39616

\[x \leq 0.5\]

Math

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

↓

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

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)
   (*
    x
    (fma
     (pow x 4.0)
     0.2
     (+
      (+ (* 0.047619047619047616 (pow x 6.0)) (* (* x x) 0.6666666666666666))
      2.0))))))

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) * (x * fma(pow(x, 4.0), 0.2, (((0.047619047619047616 * pow(x, 6.0)) + ((x * x) * 0.6666666666666666)) + 2.0)))));
}

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(x * fma((x ^ 4.0), 0.2, Float64(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(Float64(x * x) * 0.6666666666666666)) + 2.0)))))
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[(x * N[(N[Power[x, 4.0], $MachinePrecision] * 0.2 + N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.23

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

2. Simplified 0.85

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

   [Start] 0.23

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

   associate-*l/ [=>] 0.88

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

3. Applied egg-rr 0.21

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

4. Applied egg-rr 0.21

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

5. Applied egg-rr 0.21

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

6. Simplified 0.21

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

   [Start] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) + {\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
   distribute-lft-out [=>] 0.21	\[ \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + x \cdot \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}\right| \]
   distribute-lft-in [<=] 0.21	\[ \left|{\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right)}\right| \]
   +-commutative [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)}\right)\right| \]
   fma-udef [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right| \]
   associate-+l+ [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(0.2 \cdot {x}^{4} + \left(0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)}\right)\right| \]
   *-commutative [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{{x}^{4} \cdot 0.2} + \left(0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right)\right| \]
   fma-def [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{4}, 0.2, 0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)}\right)\right| \]
   fma-def [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left({x}^{4}, 0.2, \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)}\right)\right)\right| \]
   fma-udef [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left({x}^{4}, 0.2, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{0.6666666666666666 \cdot \left(x \cdot x\right) + 2}\right)\right)\right)\right| \]
   associate-*r* [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left({x}^{4}, 0.2, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{\left(0.6666666666666666 \cdot x\right) \cdot x} + 2\right)\right)\right)\right| \]
   *-commutative [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left({x}^{4}, 0.2, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{x \cdot \left(0.6666666666666666 \cdot x\right)} + 2\right)\right)\right)\right| \]
   fma-def [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left({x}^{4}, 0.2, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{\mathsf{fma}\left(x, 0.6666666666666666 \cdot x, 2\right)}\right)\right)\right)\right| \]
   *-commutative [=>] 0.21	\[ \left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left({x}^{4}, 0.2, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.6666666666666666}, 2\right)\right)\right)\right)\right| \]

7. Applied egg-rr 0.21

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

8. Final simplification 0.21

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

Alternatives

Alternative 1
Error	0.21%
Cost	39616

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

Alternative 2
Error	1.11%
Cost	32896

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

Alternative 3
Error	0.87%
Cost	32772

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \end{array} \]

Alternative 4
Error	1.11%
Cost	26372

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{0.047619047619047616}{\left|\frac{\sqrt{\pi}}{-{x}^{7}}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \end{array} \]

Alternative 5
Error	1.76%
Cost	26308

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{0.047619047619047616}{\left|\frac{\sqrt{\pi}}{-{x}^{7}}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 6
Error	1.75%
Cost	26116

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\frac{0.047619047619047616}{\left|\frac{\sqrt{\pi}}{-{x}^{7}}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 7
Error	4.9%
Cost	26052

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

Alternative 8
Error	1.75%
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;0.047619047619047616 \cdot \left|\frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 9
Error	1.74%
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 10
Error	1.74%
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\frac{{x}^{7}}{\frac{\sqrt{\pi}}{0.047619047619047616}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.6666666666666666 + 2\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 11
Error	7.26%
Cost	19840

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

Alternative 12
Error	7.15%
Cost	19456

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

Reproduce

herbie shell --seed 2023093 
(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)))))))