Average Error: 1.0 → 0.0
Time: 7.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r178293 = 4.0;
        double r178294 = 3.0;
        double r178295 = atan2(1.0, 0.0);
        double r178296 = r178294 * r178295;
        double r178297 = 1.0;
        double r178298 = v;
        double r178299 = r178298 * r178298;
        double r178300 = r178297 - r178299;
        double r178301 = r178296 * r178300;
        double r178302 = 2.0;
        double r178303 = 6.0;
        double r178304 = r178303 * r178299;
        double r178305 = r178302 - r178304;
        double r178306 = sqrt(r178305);
        double r178307 = r178301 * r178306;
        double r178308 = r178293 / r178307;
        return r178308;
}
↓
double f(double v) {
        double r178309 = 4.0;
        double r178310 = 3.0;
        double r178311 = atan2(1.0, 0.0);
        double r178312 = r178310 * r178311;
        double r178313 = 1.0;
        double r178314 = v;
        double r178315 = r178314 * r178314;
        double r178316 = r178313 - r178315;
        double r178317 = r178312 * r178316;
        double r178318 = r178309 / r178317;
        double r178319 = 2.0;
        double r178320 = 6.0;
        double r178321 = r178320 * r178315;
        double r178322 = r178319 - r178321;
        double r178323 = sqrt(r178322);
        double r178324 = r178318 / r178323;
        return r178324;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.0

    \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}\]

  4. Final simplification0.0

    \[\leadsto \frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))