Average Error: 0.5 → 0.4
Time: 29.0s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\sqrt[3]{\frac{1}{2}} \cdot \left(\frac{\cos th \cdot {a2}^{2}}{\left|\sqrt[3]{\sqrt{2}}\right|} + \frac{\left(a1 \cdot \cos th\right) \cdot a1}{\left|\sqrt[3]{\sqrt{2}}\right|}\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\sqrt[3]{\frac{1}{2}} \cdot \left(\frac{\cos th \cdot {a2}^{2}}{\left|\sqrt[3]{\sqrt{2}}\right|} + \frac{\left(a1 \cdot \cos th\right) \cdot a1}{\left|\sqrt[3]{\sqrt{2}}\right|}\right)
double f(double a1, double a2, double th) {
        double r102371 = th;
        double r102372 = cos(r102371);
        double r102373 = 2.0;
        double r102374 = sqrt(r102373);
        double r102375 = r102372 / r102374;
        double r102376 = a1;
        double r102377 = r102376 * r102376;
        double r102378 = r102375 * r102377;
        double r102379 = a2;
        double r102380 = r102379 * r102379;
        double r102381 = r102375 * r102380;
        double r102382 = r102378 + r102381;
        return r102382;
}
↓
double f(double a1, double a2, double th) {
        double r102383 = 1.0;
        double r102384 = 2.0;
        double r102385 = r102383 / r102384;
        double r102386 = cbrt(r102385);
        double r102387 = th;
        double r102388 = cos(r102387);
        double r102389 = a2;
        double r102390 = 2.0;
        double r102391 = pow(r102389, r102390);
        double r102392 = r102388 * r102391;
        double r102393 = sqrt(r102384);
        double r102394 = cbrt(r102393);
        double r102395 = fabs(r102394);
        double r102396 = r102392 / r102395;
        double r102397 = a1;
        double r102398 = r102397 * r102388;
        double r102399 = r102398 * r102397;
        double r102400 = r102399 / r102395;
        double r102401 = r102396 + r102400;
        double r102402 = r102386 * r102401;
        return r102402;
}

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

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied sqrt-prod0.5

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

  6. Applied associate-/r*0.5

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

  8. Applied add-cube-cbrt0.5

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

  9. Applied sqrt-prod0.6

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

  10. Applied div-inv0.6

    \[\leadsto \frac{\color{blue}{\cos th \cdot \frac{1}{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\]

  11. Applied times-frac0.5

    \[\leadsto \color{blue}{\left(\frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\]

  12. Simplified0.5

    \[\leadsto \left(\color{blue}{\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\]

  13. Taylor expanded around inf 0.6

    \[\leadsto \color{blue}{\frac{{a1}^{2} \cdot \cos th}{\left|{\left(\sqrt{2}\right)}^{\frac{1}{3}}\right|} \cdot {\left(\frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}^{\frac{1}{3}} + \frac{\cos th \cdot {a2}^{2}}{\left|{\left(\sqrt{2}\right)}^{\frac{1}{3}}\right|} \cdot {\left(\frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}^{\frac{1}{3}}}\]

  14. Simplified0.4

    \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{2}} \cdot \left(\frac{{a1}^{2} \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} + \frac{\cos th \cdot {a2}^{2}}{\left|\sqrt[3]{\sqrt{2}}\right|}\right)}\]
  15. Using strategy rm

  16. Applied sqr-pow0.4

    \[\leadsto \sqrt[3]{\frac{1}{2}} \cdot \left(\frac{\color{blue}{\left({a1}^{\left(\frac{2}{2}\right)} \cdot {a1}^{\left(\frac{2}{2}\right)}\right)} \cdot \cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} + \frac{\cos th \cdot {a2}^{2}}{\left|\sqrt[3]{\sqrt{2}}\right|}\right)\]

  17. Applied associate-*l*0.4

    \[\leadsto \sqrt[3]{\frac{1}{2}} \cdot \left(\frac{\color{blue}{{a1}^{\left(\frac{2}{2}\right)} \cdot \left({a1}^{\left(\frac{2}{2}\right)} \cdot \cos th\right)}}{\left|\sqrt[3]{\sqrt{2}}\right|} + \frac{\cos th \cdot {a2}^{2}}{\left|\sqrt[3]{\sqrt{2}}\right|}\right)\]

  18. Simplified0.4

    \[\leadsto \sqrt[3]{\frac{1}{2}} \cdot \left(\frac{{a1}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(a1 \cdot \cos th\right)}}{\left|\sqrt[3]{\sqrt{2}}\right|} + \frac{\cos th \cdot {a2}^{2}}{\left|\sqrt[3]{\sqrt{2}}\right|}\right)\]

  19. Final simplification0.4

    \[\leadsto \sqrt[3]{\frac{1}{2}} \cdot \left(\frac{\cos th \cdot {a2}^{2}}{\left|\sqrt[3]{\sqrt{2}}\right|} + \frac{\left(a1 \cdot \cos th\right) \cdot a1}{\left|\sqrt[3]{\sqrt{2}}\right|}\right)\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))