Average Error: 0.5 → 0.4
Time: 6.7s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\cos th \cdot \mathsf{hypot}\left(a1, a2\right)\right) \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\cos th \cdot \mathsf{hypot}\left(a1, a2\right)\right) \cdot \sqrt{\frac{a1 \cdot a1 + a2 \cdot a2}{2}}
double f(double a1, double a2, double th) {
        double r102574 = th;
        double r102575 = cos(r102574);
        double r102576 = 2.0;
        double r102577 = sqrt(r102576);
        double r102578 = r102575 / r102577;
        double r102579 = a1;
        double r102580 = r102579 * r102579;
        double r102581 = r102578 * r102580;
        double r102582 = a2;
        double r102583 = r102582 * r102582;
        double r102584 = r102578 * r102583;
        double r102585 = r102581 + r102584;
        return r102585;
}
↓
double f(double a1, double a2, double th) {
        double r102586 = th;
        double r102587 = cos(r102586);
        double r102588 = a1;
        double r102589 = a2;
        double r102590 = hypot(r102588, r102589);
        double r102591 = r102587 * r102590;
        double r102592 = r102588 * r102588;
        double r102593 = r102589 * r102589;
        double r102594 = r102592 + r102593;
        double r102595 = 2.0;
        double r102596 = r102594 / r102595;
        double r102597 = sqrt(r102596);
        double r102598 = r102591 * r102597;
        return r102598;
}

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 *-un-lft-identity0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied times-frac0.5

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

  11. Simplified0.5

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

  12. Simplified0.4

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

  14. Applied hypot-udef0.5

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

  15. Applied sqrt-undiv0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 +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))))