Average Error: 12.8 → 0.4
Time: 10.6s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{{\left(\left|w \cdot r\right|\right)}^{2}}} + 4.5\right)\]
\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{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{{\left(\left|w \cdot r\right|\right)}^{2}}} + 4.5\right)
double f(double v, double w, double r) {
        double r15018 = 3.0;
        double r15019 = 2.0;
        double r15020 = r;
        double r15021 = r15020 * r15020;
        double r15022 = r15019 / r15021;
        double r15023 = r15018 + r15022;
        double r15024 = 0.125;
        double r15025 = v;
        double r15026 = r15019 * r15025;
        double r15027 = r15018 - r15026;
        double r15028 = r15024 * r15027;
        double r15029 = w;
        double r15030 = r15029 * r15029;
        double r15031 = r15030 * r15020;
        double r15032 = r15031 * r15020;
        double r15033 = r15028 * r15032;
        double r15034 = 1.0;
        double r15035 = r15034 - r15025;
        double r15036 = r15033 / r15035;
        double r15037 = r15023 - r15036;
        double r15038 = 4.5;
        double r15039 = r15037 - r15038;
        return r15039;
}
↓
double f(double v, double w, double r) {
        double r15040 = 3.0;
        double r15041 = 2.0;
        double r15042 = r;
        double r15043 = r15042 * r15042;
        double r15044 = r15041 / r15043;
        double r15045 = r15040 + r15044;
        double r15046 = 0.125;
        double r15047 = v;
        double r15048 = r15041 * r15047;
        double r15049 = r15040 - r15048;
        double r15050 = r15046 * r15049;
        double r15051 = 1.0;
        double r15052 = r15051 - r15047;
        double r15053 = w;
        double r15054 = r15053 * r15042;
        double r15055 = fabs(r15054);
        double r15056 = 2.0;
        double r15057 = pow(r15055, r15056);
        double r15058 = r15052 / r15057;
        double r15059 = r15050 / r15058;
        double r15060 = 4.5;
        double r15061 = r15059 + r15060;
        double r15062 = r15045 - r15061;
        return r15062;
}

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 12.8

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

  2. Simplified8.4

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

  4. Applied associate-*l*2.6

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

  6. Applied add-sqr-sqrt2.6

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

  7. Simplified2.6

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

  8. Simplified0.3

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

  10. Applied fma-udef0.3

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 +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))