Average Error: 0.5 → 0.4
Time: 4.0m
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}} + \frac{a1 \cdot \cos th}{\frac{\sqrt{2}}{a1}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}} + \frac{a1 \cdot \cos th}{\frac{\sqrt{2}}{a1}}
double f(double a1, double a2, double th) {
        double r16404811 = th;
        double r16404812 = cos(r16404811);
        double r16404813 = 2.0;
        double r16404814 = sqrt(r16404813);
        double r16404815 = r16404812 / r16404814;
        double r16404816 = a1;
        double r16404817 = r16404816 * r16404816;
        double r16404818 = r16404815 * r16404817;
        double r16404819 = a2;
        double r16404820 = r16404819 * r16404819;
        double r16404821 = r16404815 * r16404820;
        double r16404822 = r16404818 + r16404821;
        return r16404822;
}
↓
double f(double a1, double a2, double th) {
        double r16404823 = a2;
        double r16404824 = th;
        double r16404825 = cos(r16404824);
        double r16404826 = r16404823 * r16404825;
        double r16404827 = 2.0;
        double r16404828 = sqrt(r16404827);
        double r16404829 = r16404826 / r16404828;
        double r16404830 = r16404823 * r16404829;
        double r16404831 = a1;
        double r16404832 = r16404831 * r16404825;
        double r16404833 = r16404828 / r16404831;
        double r16404834 = r16404832 / r16404833;
        double r16404835 = r16404830 + r16404834;
        return r16404835;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.5

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  6. Taylor expanded around inf 0.5

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

  7. Simplified0.5

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied sqrt-prod0.5

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

  12. Applied div-inv0.5

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

  13. Applied times-frac0.4

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

  14. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\frac{{a1}^{2} \cdot \cos th}{\sqrt{2}}} + a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}}\]

  15. Simplified0.4

    \[\leadsto \color{blue}{\frac{\cos th \cdot a1}{\frac{\sqrt{2}}{a1}}} + a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}}\]

  16. Final simplification0.4

    \[\leadsto a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}} + \frac{a1 \cdot \cos th}{\frac{\sqrt{2}}{a1}}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))