Average Error: 0.5 → 0.6
Time: 1.1m
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}}}{\frac{\sqrt{\sqrt{2}}}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{\sqrt{2}}}}}}\]
\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}{\sqrt{\sqrt{\sqrt{2}}}}}{\frac{\sqrt{\sqrt{2}}}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt{\sqrt{2}}}}}}
double f(double a1, double a2, double th) {
        double r300112 = th;
        double r300113 = cos(r300112);
        double r300114 = 2.0;
        double r300115 = sqrt(r300114);
        double r300116 = r300113 / r300115;
        double r300117 = a1;
        double r300118 = r300117 * r300117;
        double r300119 = r300116 * r300118;
        double r300120 = a2;
        double r300121 = r300120 * r300120;
        double r300122 = r300116 * r300121;
        double r300123 = r300119 + r300122;
        return r300123;
}
↓
double f(double a1, double a2, double th) {
        double r300124 = th;
        double r300125 = cos(r300124);
        double r300126 = 2.0;
        double r300127 = sqrt(r300126);
        double r300128 = sqrt(r300127);
        double r300129 = sqrt(r300128);
        double r300130 = r300125 / r300129;
        double r300131 = a1;
        double r300132 = a2;
        double r300133 = r300132 * r300132;
        double r300134 = fma(r300131, r300131, r300133);
        double r300135 = r300134 / r300129;
        double r300136 = r300128 / r300135;
        double r300137 = r300130 / r300136;
        return r300137;
}

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-prod0.5

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

  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-sqr-sqrt0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied times-frac0.4

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

  12. Applied associate-/l*0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2020065 +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))))