Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(1 \cdot v, v, {v}^{4}\right)\right) \cdot \left(4 \cdot \sqrt{\mathsf{fma}\left(3 \cdot \left(v \cdot v\right), \mathsf{fma}\left(3 \cdot v, v, 1\right), 1 \cdot 1\right)}\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(1 \cdot v, v, {v}^{4}\right)\right) \cdot \left(4 \cdot \sqrt{\mathsf{fma}\left(3 \cdot \left(v \cdot v\right), \mathsf{fma}\left(3 \cdot v, v, 1\right), 1 \cdot 1\right)}\right)}
double f(double v) {
        double r251090 = 2.0;
        double r251091 = sqrt(r251090);
        double r251092 = 4.0;
        double r251093 = r251091 / r251092;
        double r251094 = 1.0;
        double r251095 = 3.0;
        double r251096 = v;
        double r251097 = r251096 * r251096;
        double r251098 = r251095 * r251097;
        double r251099 = r251094 - r251098;
        double r251100 = sqrt(r251099);
        double r251101 = r251093 * r251100;
        double r251102 = r251094 - r251097;
        double r251103 = r251101 * r251102;
        return r251103;
}
↓
double f(double v) {
        double r251104 = 2.0;
        double r251105 = sqrt(r251104);
        double r251106 = 1.0;
        double r251107 = 3.0;
        double r251108 = pow(r251106, r251107);
        double r251109 = 3.0;
        double r251110 = v;
        double r251111 = r251110 * r251110;
        double r251112 = r251109 * r251111;
        double r251113 = pow(r251112, r251107);
        double r251114 = r251108 - r251113;
        double r251115 = sqrt(r251114);
        double r251116 = r251105 * r251115;
        double r251117 = pow(r251111, r251107);
        double r251118 = r251108 - r251117;
        double r251119 = r251116 * r251118;
        double r251120 = r251106 * r251110;
        double r251121 = 4.0;
        double r251122 = pow(r251110, r251121);
        double r251123 = fma(r251120, r251110, r251122);
        double r251124 = fma(r251106, r251106, r251123);
        double r251125 = 4.0;
        double r251126 = r251109 * r251110;
        double r251127 = fma(r251126, r251110, r251106);
        double r251128 = r251106 * r251106;
        double r251129 = fma(r251112, r251127, r251128);
        double r251130 = sqrt(r251129);
        double r251131 = r251125 * r251130;
        double r251132 = r251124 * r251131;
        double r251133 = r251119 / r251132;
        return r251133;
}

Error

Bits error versus v

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied flip3--0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

  4. Applied flip3--0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{\color{blue}{\frac{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}\]

  5. Applied sqrt-div0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\frac{\sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}\]

  6. Applied frac-times0.0

    \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}{4 \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}} \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}\]

  7. Applied frac-times0.0

    \[\leadsto \color{blue}{\frac{\left(\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\left(4 \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}}\]

  8. Simplified0.0

    \[\leadsto \frac{\left(\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(1 \cdot v, v, {v}^{4}\right)\right) \cdot \left(4 \cdot \sqrt{\mathsf{fma}\left(3 \cdot \left(v \cdot v\right), \mathsf{fma}\left(3 \cdot v, v, 1\right), 1 \cdot 1\right)}\right)}}\]

  9. Final simplification0.0

    \[\leadsto \frac{\left(\sqrt{2} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(1 \cdot v, v, {v}^{4}\right)\right) \cdot \left(4 \cdot \sqrt{\mathsf{fma}\left(3 \cdot \left(v \cdot v\right), \mathsf{fma}\left(3 \cdot v, v, 1\right), 1 \cdot 1\right)}\right)}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))