Average Error: 12.7 → 0.4
Time: 5.3s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right), \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
\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{2}{r \cdot r} - \left(\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right), \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 r18065 = 3.0;
        double r18066 = 2.0;
        double r18067 = r;
        double r18068 = r18067 * r18067;
        double r18069 = r18066 / r18068;
        double r18070 = r18065 + r18069;
        double r18071 = 0.125;
        double r18072 = v;
        double r18073 = r18066 * r18072;
        double r18074 = r18065 - r18073;
        double r18075 = r18071 * r18074;
        double r18076 = w;
        double r18077 = r18076 * r18076;
        double r18078 = r18077 * r18067;
        double r18079 = r18078 * r18067;
        double r18080 = r18075 * r18079;
        double r18081 = 1.0;
        double r18082 = r18081 - r18072;
        double r18083 = r18080 / r18082;
        double r18084 = r18070 - r18083;
        double r18085 = 4.5;
        double r18086 = r18084 - r18085;
        return r18086;
}
↓
double f(double v, double w, double r) {
        double r18087 = 2.0;
        double r18088 = r;
        double r18089 = r18088 * r18088;
        double r18090 = r18087 / r18089;
        double r18091 = 0.125;
        double r18092 = 3.0;
        double r18093 = v;
        double r18094 = r18087 * r18093;
        double r18095 = r18092 - r18094;
        double r18096 = r18091 * r18095;
        double r18097 = 1.0;
        double r18098 = r18097 - r18093;
        double r18099 = r18096 / r18098;
        double r18100 = log1p(r18099);
        double r18101 = expm1(r18100);
        double r18102 = w;
        double r18103 = r18102 * r18088;
        double r18104 = fabs(r18103);
        double r18105 = r18104 * r18104;
        double r18106 = 4.5;
        double r18107 = fma(r18101, r18105, r18106);
        double r18108 = r18107 - r18092;
        double r18109 = r18090 - r18108;
        return r18109;
}

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. Simplified8.7

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

  4. Applied add-sqr-sqrt8.8

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

  5. Simplified8.7

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

  6. Simplified0.4

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

  8. Applied expm1-log1p-u0.4

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right)}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]

  9. Final simplification0.4

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right)\right), \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]

Reproduce

herbie shell --seed 2019353 +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))