Average Error: 0.5 → 0.4
Time: 30.1s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}
double f(double a1, double a2, double th) {
        double r89034 = th;
        double r89035 = cos(r89034);
        double r89036 = 2.0;
        double r89037 = sqrt(r89036);
        double r89038 = r89035 / r89037;
        double r89039 = a1;
        double r89040 = r89039 * r89039;
        double r89041 = r89038 * r89040;
        double r89042 = a2;
        double r89043 = r89042 * r89042;
        double r89044 = r89038 * r89043;
        double r89045 = r89041 + r89044;
        return r89045;
}
↓
double f(double a1, double a2, double th) {
        double r89046 = a1;
        double r89047 = a2;
        double r89048 = r89047 * r89047;
        double r89049 = fma(r89046, r89046, r89048);
        double r89050 = th;
        double r89051 = cos(r89050);
        double r89052 = r89049 * r89051;
        double r89053 = 2.0;
        double r89054 = sqrt(r89053);
        double r89055 = cbrt(r89054);
        double r89056 = fabs(r89055);
        double r89057 = r89052 / r89056;
        double r89058 = 1.0;
        double r89059 = sqrt(r89054);
        double r89060 = r89058 / r89059;
        double r89061 = sqrt(r89055);
        double r89062 = r89060 / r89061;
        double r89063 = r89057 * r89062;
        return r89063;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

  1. Initial program 0.5

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}\]

  5. Applied sqrt-prod0.6

    \[\leadsto \frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}\]

  6. Applied associate-/r*0.5

    \[\leadsto \color{blue}{\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.5

    \[\leadsto \frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{2}}}}{\sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}}\]

  9. Applied sqrt-prod0.7

    \[\leadsto \frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{2}}}}{\color{blue}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}}\]

  10. Applied div-inv0.5

    \[\leadsto \frac{\color{blue}{\left(\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \frac{1}{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}}\]

  11. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}}\]

  12. Simplified0.4

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\]

  13. Final simplification0.4

    \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))