Average Error: 0.0 → 0.0
Time: 5.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)
double f(double v) {
        double r206082 = 2.0;
        double r206083 = sqrt(r206082);
        double r206084 = 4.0;
        double r206085 = r206083 / r206084;
        double r206086 = 1.0;
        double r206087 = 3.0;
        double r206088 = v;
        double r206089 = r206088 * r206088;
        double r206090 = r206087 * r206089;
        double r206091 = r206086 - r206090;
        double r206092 = sqrt(r206091);
        double r206093 = r206085 * r206092;
        double r206094 = r206086 - r206089;
        double r206095 = r206093 * r206094;
        return r206095;
}
↓
double f(double v) {
        double r206096 = 2.0;
        double r206097 = sqrt(r206096);
        double r206098 = 4.0;
        double r206099 = r206097 / r206098;
        double r206100 = 1.0;
        double r206101 = 3.0;
        double r206102 = v;
        double r206103 = r206102 * r206102;
        double r206104 = r206101 * r206103;
        double r206105 = r206100 - r206104;
        double r206106 = sqrt(r206105);
        double r206107 = r206100 - r206103;
        double r206108 = r206106 * r206107;
        double r206109 = r206099 * r206108;
        return r206109;
}

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 associate-*l*0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 +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))))