Average Error: 12.8 → 0.4
Time: 8.2s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{{\left(\left|w \cdot r\right|\right)}^{2}}} + 4.5\right)\]
\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{\frac{2}{r}}{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 r15162 = 3.0;
        double r15163 = 2.0;
        double r15164 = r;
        double r15165 = r15164 * r15164;
        double r15166 = r15163 / r15165;
        double r15167 = r15162 + r15166;
        double r15168 = 0.125;
        double r15169 = v;
        double r15170 = r15163 * r15169;
        double r15171 = r15162 - r15170;
        double r15172 = r15168 * r15171;
        double r15173 = w;
        double r15174 = r15173 * r15173;
        double r15175 = r15174 * r15164;
        double r15176 = r15175 * r15164;
        double r15177 = r15172 * r15176;
        double r15178 = 1.0;
        double r15179 = r15178 - r15169;
        double r15180 = r15177 / r15179;
        double r15181 = r15167 - r15180;
        double r15182 = 4.5;
        double r15183 = r15181 - r15182;
        return r15183;
}
↓
double f(double v, double w, double r) {
        double r15184 = 3.0;
        double r15185 = 2.0;
        double r15186 = r;
        double r15187 = r15185 / r15186;
        double r15188 = r15187 / r15186;
        double r15189 = r15184 + r15188;
        double r15190 = 0.125;
        double r15191 = v;
        double r15192 = r15185 * r15191;
        double r15193 = r15184 - r15192;
        double r15194 = r15190 * r15193;
        double r15195 = 1.0;
        double r15196 = r15195 - r15191;
        double r15197 = w;
        double r15198 = r15197 * r15186;
        double r15199 = fabs(r15198);
        double r15200 = 2.0;
        double r15201 = pow(r15199, r15200);
        double r15202 = r15196 / r15201;
        double r15203 = r15194 / r15202;
        double r15204 = 4.5;
        double r15205 = r15203 + r15204;
        double r15206 = r15189 - r15205;
        return r15206;
}

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. Simplified2.6

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

  7. Applied add-sqr-sqrt2.6

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

  8. Simplified2.6

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

  9. Simplified0.3

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

  11. Applied associate-/r*0.3

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

  13. Applied fma-udef0.3

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

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))