Average Error: 12.9 → 0.4
Time: 19.3s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{0.375 - 0.25 \cdot v}{1 - v} \cdot {\left(\left|w \cdot r\right|\right)}^{2}\right) - 4.5\]
\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
↓
\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \frac{0.375 - 0.25 \cdot v}{1 - v} \cdot {\left(\left|w \cdot r\right|\right)}^{2}\right) - 4.5
double f(double v, double w, double r) {
        double r25062 = 3.0;
        double r25063 = 2.0;
        double r25064 = r;
        double r25065 = r25064 * r25064;
        double r25066 = r25063 / r25065;
        double r25067 = r25062 + r25066;
        double r25068 = 0.125;
        double r25069 = v;
        double r25070 = r25063 * r25069;
        double r25071 = r25062 - r25070;
        double r25072 = r25068 * r25071;
        double r25073 = w;
        double r25074 = r25073 * r25073;
        double r25075 = r25074 * r25064;
        double r25076 = r25075 * r25064;
        double r25077 = r25072 * r25076;
        double r25078 = 1.0;
        double r25079 = r25078 - r25069;
        double r25080 = r25077 / r25079;
        double r25081 = r25067 - r25080;
        double r25082 = 4.5;
        double r25083 = r25081 - r25082;
        return r25083;
}
↓
double f(double v, double w, double r) {
        double r25084 = 3.0;
        double r25085 = 2.0;
        double r25086 = r;
        double r25087 = r25085 / r25086;
        double r25088 = r25087 / r25086;
        double r25089 = r25084 + r25088;
        double r25090 = 0.375;
        double r25091 = 0.25;
        double r25092 = v;
        double r25093 = r25091 * r25092;
        double r25094 = r25090 - r25093;
        double r25095 = 1.0;
        double r25096 = r25095 - r25092;
        double r25097 = r25094 / r25096;
        double r25098 = w;
        double r25099 = r25098 * r25086;
        double r25100 = fabs(r25099);
        double r25101 = 2.0;
        double r25102 = pow(r25100, r25101);
        double r25103 = r25097 * r25102;
        double r25104 = r25089 - r25103;
        double r25105 = 4.5;
        double r25106 = r25104 - r25105;
        return r25106;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.9

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

  2. Simplified8.6

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

  4. Applied add-sqr-sqrt8.7

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

  5. Simplified8.7

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

  6. Simplified0.4

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

  7. Taylor expanded around 0 0.4

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

  9. Applied fma-udef0.4

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

  10. Applied associate--r+0.4

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

  11. Simplified0.4

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

  13. Applied associate-/r*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019235 +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))