Average Error: 0.0 → 0.0
Time: 6.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r411129 = 2.0;
        double r411130 = sqrt(r411129);
        double r411131 = 4.0;
        double r411132 = r411130 / r411131;
        double r411133 = 1.0;
        double r411134 = 3.0;
        double r411135 = v;
        double r411136 = r411135 * r411135;
        double r411137 = r411134 * r411136;
        double r411138 = r411133 - r411137;
        double r411139 = sqrt(r411138);
        double r411140 = r411132 * r411139;
        double r411141 = r411133 - r411136;
        double r411142 = r411140 * r411141;
        return r411142;
}
↓
double f(double v) {
        double r411143 = 2.0;
        double r411144 = sqrt(r411143);
        double r411145 = 4.0;
        double r411146 = r411144 / r411145;
        double r411147 = 1.0;
        double r411148 = 3.0;
        double r411149 = v;
        double r411150 = r411149 * r411149;
        double r411151 = r411148 * r411150;
        double r411152 = r411147 - r411151;
        double r411153 = sqrt(r411152);
        double r411154 = log1p(r411153);
        double r411155 = expm1(r411154);
        double r411156 = r411146 * r411155;
        double r411157 = log1p(r411156);
        double r411158 = expm1(r411157);
        double r411159 = r411147 - r411150;
        double r411160 = r411158 * r411159;
        return r411160;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 expm1-log1p-u0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)\]
  4. Using strategy rm

  5. Applied expm1-log1p-u0.0

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right)\]

  6. Final simplification0.0

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{4} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)\right)\right) \cdot \left(1 - v \cdot v\right)\]

Reproduce

herbie shell --seed 2020046 +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))))