Rosa's TurbineBenchmark

Average Error: 12.4 → 0.6

Time: 1.7m

Precision: 64

Math

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

↓

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

C

double f(double v, double w, double r) {
        double r6162916 = 3.0;
        double r6162917 = 2.0;
        double r6162918 = r;
        double r6162919 = r6162918 * r6162918;
        double r6162920 = r6162917 / r6162919;
        double r6162921 = r6162916 + r6162920;
        double r6162922 = 0.125;
        double r6162923 = v;
        double r6162924 = r6162917 * r6162923;
        double r6162925 = r6162916 - r6162924;
        double r6162926 = r6162922 * r6162925;
        double r6162927 = w;
        double r6162928 = r6162927 * r6162927;
        double r6162929 = r6162928 * r6162918;
        double r6162930 = r6162929 * r6162918;
        double r6162931 = r6162926 * r6162930;
        double r6162932 = 1.0;
        double r6162933 = r6162932 - r6162923;
        double r6162934 = r6162931 / r6162933;
        double r6162935 = r6162921 - r6162934;
        double r6162936 = 4.5;
        double r6162937 = r6162935 - r6162936;
        return r6162937;
}

↓

double f(double v, double w, double r) {
        double r6162938 = 3.0;
        double r6162939 = 2.0;
        double r6162940 = r;
        double r6162941 = r6162940 * r6162940;
        double r6162942 = r6162939 / r6162941;
        double r6162943 = r6162938 + r6162942;
        double r6162944 = v;
        double r6162945 = -2.0;
        double r6162946 = fma(r6162944, r6162945, r6162938);
        double r6162947 = 1.0;
        double r6162948 = r6162947 - r6162944;
        double r6162949 = cbrt(r6162948);
        double r6162950 = 0.125;
        double r6162951 = sqrt(r6162950);
        double r6162952 = r6162949 / r6162951;
        double r6162953 = r6162946 / r6162952;
        double r6162954 = r6162951 / r6162949;
        double r6162955 = r6162954 / r6162949;
        double r6162956 = r6162953 * r6162955;
        double r6162957 = w;
        double r6162958 = r6162957 * r6162940;
        double r6162959 = r6162958 * r6162958;
        double r6162960 = 4.5;
        double r6162961 = fma(r6162956, r6162959, r6162960);
        double r6162962 = r6162943 - r6162961;
        return r6162962;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

1. Initial program 12.4

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

2. Simplified 0.4

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

4. Applied add-sqr-sqrt 0.7

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

5. Applied add-cube-cbrt 0.8

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

6. Applied times-frac 0.7

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

7. Applied *-un-lft-identity 0.7

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

8. Applied times-frac 0.6

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

9. Simplified 0.6

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

10. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019121 +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))