Average Error: 12.4 → 0.3
Time: 7.9s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\mathsf{fma}\left(-\frac{0.375 - 0.25 \cdot v}{1 - v}, {\left(\left|w \cdot r\right|\right)}^{2}, \frac{\frac{2}{r}}{r}\right) - \left(4.5 - 3\right)\]
\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
↓
\mathsf{fma}\left(-\frac{0.375 - 0.25 \cdot v}{1 - v}, {\left(\left|w \cdot r\right|\right)}^{2}, \frac{\frac{2}{r}}{r}\right) - \left(4.5 - 3\right)
double f(double v, double w, double r) {
        double r19083 = 3.0;
        double r19084 = 2.0;
        double r19085 = r;
        double r19086 = r19085 * r19085;
        double r19087 = r19084 / r19086;
        double r19088 = r19083 + r19087;
        double r19089 = 0.125;
        double r19090 = v;
        double r19091 = r19084 * r19090;
        double r19092 = r19083 - r19091;
        double r19093 = r19089 * r19092;
        double r19094 = w;
        double r19095 = r19094 * r19094;
        double r19096 = r19095 * r19085;
        double r19097 = r19096 * r19085;
        double r19098 = r19093 * r19097;
        double r19099 = 1.0;
        double r19100 = r19099 - r19090;
        double r19101 = r19098 / r19100;
        double r19102 = r19088 - r19101;
        double r19103 = 4.5;
        double r19104 = r19102 - r19103;
        return r19104;
}
↓
double f(double v, double w, double r) {
        double r19105 = 0.375;
        double r19106 = 0.25;
        double r19107 = v;
        double r19108 = r19106 * r19107;
        double r19109 = r19105 - r19108;
        double r19110 = 1.0;
        double r19111 = r19110 - r19107;
        double r19112 = r19109 / r19111;
        double r19113 = -r19112;
        double r19114 = w;
        double r19115 = r;
        double r19116 = r19114 * r19115;
        double r19117 = fabs(r19116);
        double r19118 = 2.0;
        double r19119 = pow(r19117, r19118);
        double r19120 = 2.0;
        double r19121 = r19120 / r19115;
        double r19122 = r19121 / r19115;
        double r19123 = fma(r19113, r19119, r19122);
        double r19124 = 4.5;
        double r19125 = 3.0;
        double r19126 = r19124 - r19125;
        double r19127 = r19123 - r19126;
        return r19127;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.4

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

  2. Simplified8.4

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

  4. Applied add-sqr-sqrt8.5

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

  5. Simplified8.5

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

  6. Simplified0.3

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

  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{\color{blue}{0.375 - 0.25 \cdot v}}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
  8. Using strategy rm

  9. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} - \left(\mathsf{fma}\left(\frac{0.375 - 0.25 \cdot v}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
  10. Using strategy rm

  11. Applied fma-udef0.3

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\color{blue}{\left(\frac{0.375 - 0.25 \cdot v}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + 4.5\right)} - 3\right)\]

  12. Applied associate--l+0.3

    \[\leadsto \frac{\frac{2}{r}}{r} - \color{blue}{\left(\frac{0.375 - 0.25 \cdot v}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)}\]

  13. Applied associate--r+0.3

    \[\leadsto \color{blue}{\left(\frac{\frac{2}{r}}{r} - \frac{0.375 - 0.25 \cdot v}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right)\right) - \left(4.5 - 3\right)}\]

  14. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(-\frac{0.375 - 0.25 \cdot v}{1 - v}, {\left(\left|w \cdot r\right|\right)}^{2}, \frac{\frac{2}{r}}{r}\right)} - \left(4.5 - 3\right)\]

  15. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(-\frac{0.375 - 0.25 \cdot v}{1 - v}, {\left(\left|w \cdot r\right|\right)}^{2}, \frac{\frac{2}{r}}{r}\right) - \left(4.5 - 3\right)\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))