Average Error: 0.0 → 0.0
Time: 18.8s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)
double f(double v) {
        double r182078 = 2.0;
        double r182079 = sqrt(r182078);
        double r182080 = 4.0;
        double r182081 = r182079 / r182080;
        double r182082 = 1.0;
        double r182083 = 3.0;
        double r182084 = v;
        double r182085 = r182084 * r182084;
        double r182086 = r182083 * r182085;
        double r182087 = r182082 - r182086;
        double r182088 = sqrt(r182087);
        double r182089 = r182081 * r182088;
        double r182090 = r182082 - r182085;
        double r182091 = r182089 * r182090;
        return r182091;
}
↓
double f(double v) {
        double r182092 = 1.0;
        double r182093 = v;
        double r182094 = r182093 * r182093;
        double r182095 = r182092 - r182094;
        double r182096 = 2.0;
        double r182097 = sqrt(r182096);
        double r182098 = 4.0;
        double r182099 = r182097 / r182098;
        double r182100 = 3.0;
        double r182101 = r182100 * r182094;
        double r182102 = r182092 - r182101;
        double r182103 = sqrt(r182102);
        double r182104 = r182099 * r182103;
        double r182105 = r182095 * r182104;
        return r182105;
}

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 sub-neg0.0

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

  4. Applied distribute-lft-in0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019294 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))