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

Average Error: 0.1 → 0.1

Time: 1.6min

Precision: binary64

Cost: 39808

\[x \leq 0.5\]

Math

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

↓

\[\left|\left(\frac{-1}{\sqrt{\pi}} \cdot \left(-\left|x\right|\right)\right) \cdot \left({x}^{4} \cdot \left(x \cdot \left(x \cdot 0.047619047619047616\right) + 0.2\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

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
  (*
   (* (/ -1.0 (sqrt PI)) (- (fabs x)))
   (+
    (* (pow x 4.0) (+ (* x (* x 0.047619047619047616)) 0.2))
    (fma 0.6666666666666666 (* x x) 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((((-1.0 / sqrt(((double) M_PI))) * -fabs(x)) * ((pow(x, 4.0) * ((x * (x * 0.047619047619047616)) + 0.2)) + fma(0.6666666666666666, (x * x), 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(Float64(Float64(-1.0 / sqrt(pi)) * Float64(-abs(x))) * Float64(Float64((x ^ 4.0) * Float64(Float64(x * Float64(x * 0.047619047619047616)) + 0.2)) + fma(0.6666666666666666, Float64(x * x), 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[(N[(-1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-N[Abs[x], $MachinePrecision])), $MachinePrecision] * N[(N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(x * N[(x * 0.047619047619047616), $MachinePrecision]), $MachinePrecision] + 0.2), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.6

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

3. Applied egg-rr 0.6

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

4. Applied egg-rr 0.1

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

5. Simplified 0.5

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

6. Applied egg-rr 0.1

   \[\leadsto \left|\color{blue}{\left(\frac{-1}{\sqrt{\pi}} \cdot \left(-\left|x\right|\right)\right)} \cdot \left({x}^{4} \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

7. Applied egg-rr 0.1

   \[\leadsto \left|\left(\frac{-1}{\sqrt{\pi}} \cdot \left(-\left|x\right|\right)\right) \cdot \left({x}^{4} \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.047619047619047616\right) + 0.2\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

Alternatives

Alternative 1
Error	0.5
Cost	39616

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

Alternative 2
Error	4.2
Cost	32896

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

Alternative 3
Error	4.4
Cost	32704

\[\left|\left(\left|x\right| \cdot \left(-\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right)\right)\right) \cdot \frac{-1}{\sqrt{\pi}}\right| \]

Alternative 4
Error	4.4
Cost	26368

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

Alternative 5
Error	4.4
Cost	26240

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

Alternative 6
Error	4.7
Cost	25856

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

Reproduce

herbie shell --seed 2023033 
(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)))))))