Average Error: 12.8 → 0.5
Time: 28.2s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} + \left(3 - \mathsf{fma}\left(\left(\frac{\sqrt{0.125}}{1 - v} \cdot \left(w \cdot r\right)\right) \cdot \frac{\sqrt{0.125}}{\frac{1}{w \cdot r}}, 3 - 2 \cdot v, 4.5\right)\right)\]
\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
↓
\frac{\frac{2}{r}}{r} + \left(3 - \mathsf{fma}\left(\left(\frac{\sqrt{0.125}}{1 - v} \cdot \left(w \cdot r\right)\right) \cdot \frac{\sqrt{0.125}}{\frac{1}{w \cdot r}}, 3 - 2 \cdot v, 4.5\right)\right)
double f(double v, double w, double r) {
        double r1333085 = 3.0;
        double r1333086 = 2.0;
        double r1333087 = r;
        double r1333088 = r1333087 * r1333087;
        double r1333089 = r1333086 / r1333088;
        double r1333090 = r1333085 + r1333089;
        double r1333091 = 0.125;
        double r1333092 = v;
        double r1333093 = r1333086 * r1333092;
        double r1333094 = r1333085 - r1333093;
        double r1333095 = r1333091 * r1333094;
        double r1333096 = w;
        double r1333097 = r1333096 * r1333096;
        double r1333098 = r1333097 * r1333087;
        double r1333099 = r1333098 * r1333087;
        double r1333100 = r1333095 * r1333099;
        double r1333101 = 1.0;
        double r1333102 = r1333101 - r1333092;
        double r1333103 = r1333100 / r1333102;
        double r1333104 = r1333090 - r1333103;
        double r1333105 = 4.5;
        double r1333106 = r1333104 - r1333105;
        return r1333106;
}
↓
double f(double v, double w, double r) {
        double r1333107 = 2.0;
        double r1333108 = r;
        double r1333109 = r1333107 / r1333108;
        double r1333110 = r1333109 / r1333108;
        double r1333111 = 3.0;
        double r1333112 = 0.125;
        double r1333113 = sqrt(r1333112);
        double r1333114 = 1.0;
        double r1333115 = v;
        double r1333116 = r1333114 - r1333115;
        double r1333117 = r1333113 / r1333116;
        double r1333118 = w;
        double r1333119 = r1333118 * r1333108;
        double r1333120 = r1333117 * r1333119;
        double r1333121 = 1.0;
        double r1333122 = r1333121 / r1333119;
        double r1333123 = r1333113 / r1333122;
        double r1333124 = r1333120 * r1333123;
        double r1333125 = r1333107 * r1333115;
        double r1333126 = r1333111 - r1333125;
        double r1333127 = 4.5;
        double r1333128 = fma(r1333124, r1333126, r1333127);
        double r1333129 = r1333111 - r1333128;
        double r1333130 = r1333110 + r1333129;
        return r1333130;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.8

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

  2. Simplified0.4

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

  4. Applied associate-/r*0.4

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

  6. Applied *-un-lft-identity0.4

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

  7. Applied times-frac0.4

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

  8. Applied *-un-lft-identity0.4

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

  9. Applied times-frac0.4

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

  10. Simplified0.4

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

  12. Applied div-inv0.4

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

  13. Applied add-sqr-sqrt0.6

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

  14. Applied times-frac0.5

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

  15. Applied associate-*r*0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))