Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B

Average Error: 0.0 → 0.0

Time: 16.3s

Precision: 64

Math

\[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]

↓

\[\left(\left(0.27061000000000002 \cdot x\right) \cdot \frac{0.707110000000000016}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + 0.707110000000000016 \cdot \left(\frac{2.30753}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} - x\right)\right) + \left(\left(-x\right) + x\right) \cdot 0.707110000000000016\]

C

double f(double x) {
        double r73705 = 0.70711;
        double r73706 = 2.30753;
        double r73707 = x;
        double r73708 = 0.27061;
        double r73709 = r73707 * r73708;
        double r73710 = r73706 + r73709;
        double r73711 = 1.0;
        double r73712 = 0.99229;
        double r73713 = 0.04481;
        double r73714 = r73707 * r73713;
        double r73715 = r73712 + r73714;
        double r73716 = r73707 * r73715;
        double r73717 = r73711 + r73716;
        double r73718 = r73710 / r73717;
        double r73719 = r73718 - r73707;
        double r73720 = r73705 * r73719;
        return r73720;
}

↓

double f(double x) {
        double r73721 = 0.27061;
        double r73722 = x;
        double r73723 = r73721 * r73722;
        double r73724 = 0.70711;
        double r73725 = 0.04481;
        double r73726 = 0.99229;
        double r73727 = fma(r73725, r73722, r73726);
        double r73728 = 1.0;
        double r73729 = fma(r73722, r73727, r73728);
        double r73730 = r73724 / r73729;
        double r73731 = r73723 * r73730;
        double r73732 = 2.30753;
        double r73733 = r73732 / r73729;
        double r73734 = r73733 - r73722;
        double r73735 = r73724 * r73734;
        double r73736 = r73731 + r73735;
        double r73737 = -r73722;
        double r73738 = r73737 + r73722;
        double r73739 = r73738 * r73724;
        double r73740 = r73736 + r73739;
        return r73740;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

   \[0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - x\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.6

   \[\leadsto 0.707110000000000016 \cdot \left(\frac{2.30753 + x \cdot 0.27061000000000002}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)} - \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)\]

4. Applied div-inv 0.6

   \[\leadsto 0.707110000000000016 \cdot \left(\color{blue}{\left(2.30753 + x \cdot 0.27061000000000002\right) \cdot \frac{1}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)\]

5. Applied prod-diff 0.6

   \[\leadsto 0.707110000000000016 \cdot \color{blue}{\left(\mathsf{fma}\left(2.30753 + x \cdot 0.27061000000000002, \frac{1}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)}\]

6. Applied distribute-lft-in 0.6

   \[\leadsto \color{blue}{0.707110000000000016 \cdot \mathsf{fma}\left(2.30753 + x \cdot 0.27061000000000002, \frac{1}{1 + x \cdot \left(0.992290000000000005 + x \cdot 0.044810000000000003\right)}, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + 0.707110000000000016 \cdot \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}\]

7. Simplified 0.1

   \[\leadsto \color{blue}{0.707110000000000016 \cdot \left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) - x\right)} + 0.707110000000000016 \cdot \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\]

8. Simplified 0.0

   \[\leadsto 0.707110000000000016 \cdot \left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \mathsf{fma}\left(0.27061000000000002, x, 2.30753\right) - x\right) + \color{blue}{\left(\left(-x\right) + x\right) \cdot 0.707110000000000016}\]
9. Using strategy rm

10. Applied fma-udef 0.0

    \[\leadsto 0.707110000000000016 \cdot \left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \color{blue}{\left(0.27061000000000002 \cdot x + 2.30753\right)} - x\right) + \left(\left(-x\right) + x\right) \cdot 0.707110000000000016\]

11. Applied distribute-lft-in 0.0

    \[\leadsto 0.707110000000000016 \cdot \left(\color{blue}{\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \left(0.27061000000000002 \cdot x\right) + \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot 2.30753\right)} - x\right) + \left(\left(-x\right) + x\right) \cdot 0.707110000000000016\]

12. Applied associate--l+ 0.0

    \[\leadsto 0.707110000000000016 \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \left(0.27061000000000002 \cdot x\right) + \left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot 2.30753 - x\right)\right)} + \left(\left(-x\right) + x\right) \cdot 0.707110000000000016\]

13. Applied distribute-lft-in 0.0

    \[\leadsto \color{blue}{\left(0.707110000000000016 \cdot \left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot \left(0.27061000000000002 \cdot x\right)\right) + 0.707110000000000016 \cdot \left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot 2.30753 - x\right)\right)} + \left(\left(-x\right) + x\right) \cdot 0.707110000000000016\]

14. Simplified 0.0

    \[\leadsto \left(\color{blue}{\left(0.27061000000000002 \cdot x\right) \cdot \frac{0.707110000000000016}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)}} + 0.707110000000000016 \cdot \left(\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} \cdot 2.30753 - x\right)\right) + \left(\left(-x\right) + x\right) \cdot 0.707110000000000016\]

15. Simplified 0.0

    \[\leadsto \left(\left(0.27061000000000002 \cdot x\right) \cdot \frac{0.707110000000000016}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + \color{blue}{0.707110000000000016 \cdot \left(\frac{2.30753}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} - x\right)}\right) + \left(\left(-x\right) + x\right) \cdot 0.707110000000000016\]

16. Final simplification 0.0

    \[\leadsto \left(\left(0.27061000000000002 \cdot x\right) \cdot \frac{0.707110000000000016}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} + 0.707110000000000016 \cdot \left(\frac{2.30753}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.044810000000000003, x, 0.992290000000000005\right), 1\right)} - x\right)\right) + \left(\left(-x\right) + x\right) \cdot 0.707110000000000016\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))