Average Error: 0.5 → 0.4
Time: 7.9s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th \cdot \mathsf{hypot}\left(a1, a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\mathsf{hypot}\left(a1, a2\right)}{\sqrt[3]{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\cos th \cdot \mathsf{hypot}\left(a1, a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\mathsf{hypot}\left(a1, a2\right)}{\sqrt[3]{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r103373 = th;
        double r103374 = cos(r103373);
        double r103375 = 2.0;
        double r103376 = sqrt(r103375);
        double r103377 = r103374 / r103376;
        double r103378 = a1;
        double r103379 = r103378 * r103378;
        double r103380 = r103377 * r103379;
        double r103381 = a2;
        double r103382 = r103381 * r103381;
        double r103383 = r103377 * r103382;
        double r103384 = r103380 + r103383;
        return r103384;
}
↓
double f(double a1, double a2, double th) {
        double r103385 = th;
        double r103386 = cos(r103385);
        double r103387 = a1;
        double r103388 = a2;
        double r103389 = hypot(r103387, r103388);
        double r103390 = r103386 * r103389;
        double r103391 = 2.0;
        double r103392 = sqrt(r103391);
        double r103393 = cbrt(r103392);
        double r103394 = r103393 * r103393;
        double r103395 = r103390 / r103394;
        double r103396 = r103389 / r103393;
        double r103397 = r103395 * r103396;
        return r103397;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  5. Applied associate-*r*0.5

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

  6. Simplified0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied times-frac0.5

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019353 +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))))