Average Error: 0.5 → 0.5
Time: 1.2m
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[a1 \cdot \left(\frac{\cos th}{\sqrt{2}} \cdot a1\right) + \left(\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a2 \cdot a2\right)\]
double f(double a1, double a2, double th) {
        double r6630642 = th;
        double r6630643 = cos(r6630642);
        double r6630644 = 2.0;
        double r6630645 = sqrt(r6630644);
        double r6630646 = r6630643 / r6630645;
        double r6630647 = a1;
        double r6630648 = r6630647 * r6630647;
        double r6630649 = r6630646 * r6630648;
        double r6630650 = a2;
        double r6630651 = r6630650 * r6630650;
        double r6630652 = r6630646 * r6630651;
        double r6630653 = r6630649 + r6630652;
        return r6630653;
}
↓
double f(double a1, double a2, double th) {
        double r6630654 = a1;
        double r6630655 = th;
        double r6630656 = cos(r6630655);
        double r6630657 = 2.0;
        double r6630658 = sqrt(r6630657);
        double r6630659 = r6630656 / r6630658;
        double r6630660 = r6630659 * r6630654;
        double r6630661 = r6630654 * r6630660;
        double r6630662 = cbrt(r6630658);
        double r6630663 = fabs(r6630662);
        double r6630664 = r6630656 / r6630663;
        double r6630665 = 1.0;
        double r6630666 = sqrt(r6630658);
        double r6630667 = r6630665 / r6630666;
        double r6630668 = sqrt(r6630662);
        double r6630669 = r6630667 / r6630668;
        double r6630670 = r6630664 * r6630669;
        double r6630671 = a2;
        double r6630672 = r6630671 * r6630671;
        double r6630673 = r6630670 * r6630672;
        double r6630674 = r6630661 + r6630673;
        return r6630674;
}


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

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. Using strategy rm

  3. Applied associate-*r*0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied sqrt-prod0.5

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

  7. Applied associate-/r*0.5

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

  9. Applied add-cube-cbrt0.5

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

  10. Applied sqrt-prod0.6

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

  11. Applied div-inv0.5

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

  12. Applied times-frac0.5

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

  13. Simplified0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2019101 +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))))