Average Error: 12.5 → 0.4
Time: 26.6s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v} \cdot \mathsf{fma}\left(-2, v, 3\right) + 4.5\right)\]
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v} \cdot \mathsf{fma}\left(-2, v, 3\right) + 4.5\right)
double f(double v, double w, double r) {
        double r595026 = 3.0;
        double r595027 = 2.0;
        double r595028 = r;
        double r595029 = r595028 * r595028;
        double r595030 = r595027 / r595029;
        double r595031 = r595026 + r595030;
        double r595032 = 0.125;
        double r595033 = v;
        double r595034 = r595027 * r595033;
        double r595035 = r595026 - r595034;
        double r595036 = r595032 * r595035;
        double r595037 = w;
        double r595038 = r595037 * r595037;
        double r595039 = r595038 * r595028;
        double r595040 = r595039 * r595028;
        double r595041 = r595036 * r595040;
        double r595042 = 1.0;
        double r595043 = r595042 - r595033;
        double r595044 = r595041 / r595043;
        double r595045 = r595031 - r595044;
        double r595046 = 4.5;
        double r595047 = r595045 - r595046;
        return r595047;
}
↓
double f(double v, double w, double r) {
        double r595048 = 3.0;
        double r595049 = 2.0;
        double r595050 = r;
        double r595051 = r595050 * r595050;
        double r595052 = r595049 / r595051;
        double r595053 = r595048 + r595052;
        double r595054 = 0.125;
        double r595055 = w;
        double r595056 = r595055 * r595050;
        double r595057 = r595054 * r595056;
        double r595058 = r595057 * r595056;
        double r595059 = 1.0;
        double r595060 = v;
        double r595061 = r595059 - r595060;
        double r595062 = r595058 / r595061;
        double r595063 = -2.0;
        double r595064 = fma(r595063, r595060, r595048);
        double r595065 = r595062 * r595064;
        double r595066 = 4.5;
        double r595067 = r595065 + r595066;
        double r595068 = r595053 - r595067;
        return r595068;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.5

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\frac{0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}, \left(w \cdot r\right) \cdot \left(w \cdot r\right), 4.5\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) + 4.5\right)}\]
  5. Using strategy rm

  6. Applied associate-*r*0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{0.125}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}} \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)} + 4.5\right)\]
  7. Using strategy rm

  8. Applied associate-*l/0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{0.125 \cdot \left(w \cdot r\right)}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}} \cdot \left(w \cdot r\right) + 4.5\right)\]

  9. Applied associate-*l/0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{\frac{1 - v}{\mathsf{fma}\left(-2, v, 3\right)}}} + 4.5\right)\]
  10. Using strategy rm

  11. Applied associate-/r/0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v} \cdot \mathsf{fma}\left(-2, v, 3\right)} + 4.5\right)\]

  12. Final simplification0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}{1 - v} \cdot \mathsf{fma}\left(-2, v, 3\right) + 4.5\right)\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))