Average Error: 12.7 → 0.3
Time: 5.3s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
\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
↓
\frac{\frac{2}{r}}{r} - \left(\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) - 3\right)
double f(double v, double w, double r) {
        double r15111 = 3.0;
        double r15112 = 2.0;
        double r15113 = r;
        double r15114 = r15113 * r15113;
        double r15115 = r15112 / r15114;
        double r15116 = r15111 + r15115;
        double r15117 = 0.125;
        double r15118 = v;
        double r15119 = r15112 * r15118;
        double r15120 = r15111 - r15119;
        double r15121 = r15117 * r15120;
        double r15122 = w;
        double r15123 = r15122 * r15122;
        double r15124 = r15123 * r15113;
        double r15125 = r15124 * r15113;
        double r15126 = r15121 * r15125;
        double r15127 = 1.0;
        double r15128 = r15127 - r15118;
        double r15129 = r15126 / r15128;
        double r15130 = r15116 - r15129;
        double r15131 = 4.5;
        double r15132 = r15130 - r15131;
        return r15132;
}
↓
double f(double v, double w, double r) {
        double r15133 = 2.0;
        double r15134 = r;
        double r15135 = r15133 / r15134;
        double r15136 = r15135 / r15134;
        double r15137 = 0.125;
        double r15138 = 3.0;
        double r15139 = v;
        double r15140 = r15133 * r15139;
        double r15141 = r15138 - r15140;
        double r15142 = r15137 * r15141;
        double r15143 = 1.0;
        double r15144 = r15143 - r15139;
        double r15145 = r15142 / r15144;
        double r15146 = w;
        double r15147 = r15146 * r15134;
        double r15148 = fabs(r15147);
        double r15149 = r15148 * r15148;
        double r15150 = 4.5;
        double r15151 = fma(r15145, r15149, r15150);
        double r15152 = r15151 - r15138;
        double r15153 = r15136 - r15152;
        return r15153;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.7

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

  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-sqrt8.4

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

  5. Simplified8.4

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

  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. Using strategy rm

  8. Applied associate-/r*0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020021 +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))