Average Error: 0.1 → 0.2
Time: 26.1s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot e\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot e
double f(double e, double v) {
        double r815105 = e;
        double r815106 = v;
        double r815107 = sin(r815106);
        double r815108 = r815105 * r815107;
        double r815109 = 1.0;
        double r815110 = cos(r815106);
        double r815111 = r815105 * r815110;
        double r815112 = r815109 + r815111;
        double r815113 = r815108 / r815112;
        return r815113;
}
↓
double f(double e, double v) {
        double r815114 = v;
        double r815115 = sin(r815114);
        double r815116 = cos(r815114);
        double r815117 = e;
        double r815118 = 1.0;
        double r815119 = fma(r815116, r815117, r815118);
        double r815120 = sqrt(r815119);
        double r815121 = r815115 / r815120;
        double r815122 = r815121 / r815120;
        double r815123 = r815122 * r815117;
        return r815123;
}

Error

Bits error versus e

Bits error versus v

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot e}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{\sin v}{\color{blue}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}} \cdot e\]

  5. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}} \cdot e\]

  6. Final simplification0.2

    \[\leadsto \frac{\frac{\sin v}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}}}{\sqrt{\mathsf{fma}\left(\cos v, e, 1\right)}} \cdot e\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))