Average Error: 0.1 → 0.1
Time: 3.3s
Precision: binary64
\[x \leq 0.5\]
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
↓
\[\begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ t_1 := {\left(\left|x\right|\right)}^{3}\\ t_2 := t_1 \cdot \left(x \cdot x\right)\\ \left|t_0 \cdot \left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot t_1\right) + 0.2 \cdot t_2\right) + t_0 \cdot \left(0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot t_2\right)\right)\right| \end{array} \]
(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
 (let* ((t_0 (/ 1.0 (sqrt PI))) (t_1 (pow (fabs x) 3.0)) (t_2 (* t_1 (* x x))))
   (fabs
    (+
     (* t_0 (+ (fma 2.0 (fabs x) (* 0.6666666666666666 t_1)) (* 0.2 t_2)))
     (* t_0 (* 0.047619047619047616 (* (* x x) t_2)))))))
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) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double t_1 = pow(fabs(x), 3.0);
	double t_2 = t_1 * (x * x);
	return fabs(((t_0 * (fma(2.0, fabs(x), (0.6666666666666666 * t_1)) + (0.2 * t_2))) + (t_0 * (0.047619047619047616 * ((x * x) * t_2)))));
}

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)
	t_0 = Float64(1.0 / sqrt(pi))
	t_1 = abs(x) ^ 3.0
	t_2 = Float64(t_1 * Float64(x * x))
	return abs(Float64(Float64(t_0 * Float64(fma(2.0, abs(x), Float64(0.6666666666666666 * t_1)) + Float64(0.2 * t_2))) + Float64(t_0 * Float64(0.047619047619047616 * Float64(Float64(x * x) * t_2)))))
end

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_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(t$95$0 * N[(N[(2.0 * N[Abs[x], $MachinePrecision] + N[(0.6666666666666666 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

\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|
↓
\begin{array}{l}
t_0 := \frac{1}{\sqrt{\pi}}\\
t_1 := {\left(\left|x\right|\right)}^{3}\\
t_2 := t_1 \cdot \left(x \cdot x\right)\\
\left|t_0 \cdot \left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot t_1\right) + 0.2 \cdot t_2\right) + t_0 \cdot \left(0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot t_2\right)\right)\right|
\end{array}

Error

Bits error versus x

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. Applied egg-rr0.1

    \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) + 0.2 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left({\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}\right| \]

  3. Final simplification0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) + 0.2 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

Reproduce

herbie shell --seed 2022178 
(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)))))))