Average Error: 0.5 → 0.4
Time: 1.3m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}{\sqrt{2}} \cdot \frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{t}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}{1 - v \cdot v}\]
double f(double v, double t) {
        double r31435425 = 1.0;
        double r31435426 = 5.0;
        double r31435427 = v;
        double r31435428 = r31435427 * r31435427;
        double r31435429 = r31435426 * r31435428;
        double r31435430 = r31435425 - r31435429;
        double r31435431 = atan2(1.0, 0.0);
        double r31435432 = t;
        double r31435433 = r31435431 * r31435432;
        double r31435434 = 2.0;
        double r31435435 = 3.0;
        double r31435436 = r31435435 * r31435428;
        double r31435437 = r31435425 - r31435436;
        double r31435438 = r31435434 * r31435437;
        double r31435439 = sqrt(r31435438);
        double r31435440 = r31435433 * r31435439;
        double r31435441 = r31435425 - r31435428;
        double r31435442 = r31435440 * r31435441;
        double r31435443 = r31435430 / r31435442;
        return r31435443;
}
↓
double f(double v, double t) {
        double r31435444 = 1.0;
        double r31435445 = v;
        double r31435446 = r31435445 * r31435445;
        double r31435447 = 5.0;
        double r31435448 = r31435446 * r31435447;
        double r31435449 = r31435444 - r31435448;
        double r31435450 = sqrt(r31435449);
        double r31435451 = atan2(1.0, 0.0);
        double r31435452 = r31435450 / r31435451;
        double r31435453 = 2.0;
        double r31435454 = sqrt(r31435453);
        double r31435455 = r31435452 / r31435454;
        double r31435456 = t;
        double r31435457 = r31435450 / r31435456;
        double r31435458 = 3.0;
        double r31435459 = r31435458 * r31435446;
        double r31435460 = r31435444 - r31435459;
        double r31435461 = sqrt(r31435460);
        double r31435462 = r31435457 / r31435461;
        double r31435463 = r31435455 * r31435462;
        double r31435464 = r31435444 - r31435446;
        double r31435465 = r31435463 / r31435464;
        return r31435465;
}


\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{\frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}{\sqrt{2}} \cdot \frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{t}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}{1 - v \cdot v}

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
  2. Using strategy rm

  3. Applied associate-/r*0.5

    \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}}\]
  4. Using strategy rm

  5. Applied associate-/r*0.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}}{1 - v \cdot v}\]
  6. Using strategy rm

  7. Applied sqrt-prod0.4

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\color{blue}{\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}}{1 - v \cdot v}\]

  8. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}{\pi \cdot t}}{\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}{1 - v \cdot v}\]

  9. Applied times-frac0.4

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}}}{\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}{1 - v \cdot v}\]

  10. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{2}} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}}{1 - v \cdot v}\]

  11. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{\pi}}{\sqrt{2}} \cdot \frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{t}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}{1 - v \cdot v}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))