Average Error: 12.1 → 0.7
Time: 23.1s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\frac{2}{r \cdot r} - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \left(r \cdot w\right) + 4.5\right)\right) + 3\]
\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(\frac{2}{r \cdot r} - \left(\left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \left(r \cdot w\right) + 4.5\right)\right) + 3
double f(double v, double w, double r) {
        double r1514106 = 3.0;
        double r1514107 = 2.0;
        double r1514108 = r;
        double r1514109 = r1514108 * r1514108;
        double r1514110 = r1514107 / r1514109;
        double r1514111 = r1514106 + r1514110;
        double r1514112 = 0.125;
        double r1514113 = v;
        double r1514114 = r1514107 * r1514113;
        double r1514115 = r1514106 - r1514114;
        double r1514116 = r1514112 * r1514115;
        double r1514117 = w;
        double r1514118 = r1514117 * r1514117;
        double r1514119 = r1514118 * r1514108;
        double r1514120 = r1514119 * r1514108;
        double r1514121 = r1514116 * r1514120;
        double r1514122 = 1.0;
        double r1514123 = r1514122 - r1514113;
        double r1514124 = r1514121 / r1514123;
        double r1514125 = r1514111 - r1514124;
        double r1514126 = 4.5;
        double r1514127 = r1514125 - r1514126;
        return r1514127;
}
↓
double f(double v, double w, double r) {
        double r1514128 = 2.0;
        double r1514129 = r;
        double r1514130 = r1514129 * r1514129;
        double r1514131 = r1514128 / r1514130;
        double r1514132 = 0.125;
        double r1514133 = v;
        double r1514134 = -2.0;
        double r1514135 = 3.0;
        double r1514136 = fma(r1514133, r1514134, r1514135);
        double r1514137 = r1514132 * r1514136;
        double r1514138 = w;
        double r1514139 = 1.0;
        double r1514140 = r1514139 - r1514133;
        double r1514141 = r1514129 / r1514140;
        double r1514142 = r1514138 * r1514141;
        double r1514143 = r1514137 * r1514142;
        double r1514144 = r1514129 * r1514138;
        double r1514145 = r1514143 * r1514144;
        double r1514146 = 4.5;
        double r1514147 = r1514145 + r1514146;
        double r1514148 = r1514131 - r1514147;
        double r1514149 = r1514148 + r1514135;
        return r1514149;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.1

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

  2. Simplified0.4

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

  4. Applied associate-/l*0.4

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

  6. Applied div-inv0.4

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

  7. Applied associate-/r*0.4

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

  9. Applied fma-udef0.4

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

  10. Simplified0.3

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

  12. Applied *-un-lft-identity0.3

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

  13. Applied times-frac0.7

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

  14. Simplified0.7

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

  15. Final simplification0.7

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))