Average Error: 0.5 → 0.4
Time: 4.2m
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(a1 \cdot a1\right) + \cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \left(a1 \cdot a1\right) + \cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}
double f(double a1, double a2, double th) {
        double r3959761 = th;
        double r3959762 = cos(r3959761);
        double r3959763 = 2.0;
        double r3959764 = sqrt(r3959763);
        double r3959765 = r3959762 / r3959764;
        double r3959766 = a1;
        double r3959767 = r3959766 * r3959766;
        double r3959768 = r3959765 * r3959767;
        double r3959769 = a2;
        double r3959770 = r3959769 * r3959769;
        double r3959771 = r3959765 * r3959770;
        double r3959772 = r3959768 + r3959771;
        return r3959772;
}
↓
double f(double a1, double a2, double th) {
        double r3959773 = th;
        double r3959774 = cos(r3959773);
        double r3959775 = 2.0;
        double r3959776 = sqrt(r3959775);
        double r3959777 = sqrt(r3959776);
        double r3959778 = r3959774 / r3959777;
        double r3959779 = sqrt(r3959777);
        double r3959780 = r3959778 / r3959779;
        double r3959781 = r3959780 / r3959779;
        double r3959782 = a1;
        double r3959783 = r3959782 * r3959782;
        double r3959784 = r3959781 * r3959783;
        double r3959785 = a2;
        double r3959786 = r3959776 / r3959785;
        double r3959787 = r3959785 / r3959786;
        double r3959788 = r3959774 * r3959787;
        double r3959789 = r3959784 + r3959788;
        return r3959789;
}

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 div-inv0.5

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

  4. Applied associate-*l*0.5

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

  5. Simplified0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied associate-/r*0.5

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

  11. Applied add-sqr-sqrt0.5

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

  12. Applied sqrt-prod0.5

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

  13. Applied associate-/r*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))