Average Error: 0.5 → 0.5
Time: 16.5s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r145946 = th;
        double r145947 = cos(r145946);
        double r145948 = 2.0;
        double r145949 = sqrt(r145948);
        double r145950 = r145947 / r145949;
        double r145951 = a1;
        double r145952 = r145951 * r145951;
        double r145953 = r145950 * r145952;
        double r145954 = a2;
        double r145955 = r145954 * r145954;
        double r145956 = r145950 * r145955;
        double r145957 = r145953 + r145956;
        return r145957;
}
↓
double f(double a1, double a2, double th) {
        double r145958 = th;
        double r145959 = cos(r145958);
        double r145960 = a1;
        double r145961 = a2;
        double r145962 = r145961 * r145961;
        double r145963 = fma(r145960, r145960, r145962);
        double r145964 = r145959 * r145963;
        double r145965 = 2.0;
        double r145966 = sqrt(r145965);
        double r145967 = cbrt(r145966);
        double r145968 = r145967 * r145967;
        double r145969 = r145964 / r145968;
        double r145970 = r145969 / r145967;
        return r145970;
}

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-cube-cbrt0.5

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

  5. Applied associate-/r*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019344 +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))))