Average Error: 0.5 → 0.5
Time: 26.0s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\cos th \cdot \left(\frac{\frac{a2 \cdot a2}{\sqrt{\sqrt{\sqrt[3]{2}}} \cdot \sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}} + \frac{a1 \cdot a1}{\sqrt{2}}\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\cos th \cdot \left(\frac{\frac{a2 \cdot a2}{\sqrt{\sqrt{\sqrt[3]{2}}} \cdot \sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}} + \frac{a1 \cdot a1}{\sqrt{2}}\right)
double f(double a1, double a2, double th) {
        double r74258 = th;
        double r74259 = cos(r74258);
        double r74260 = 2.0;
        double r74261 = sqrt(r74260);
        double r74262 = r74259 / r74261;
        double r74263 = a1;
        double r74264 = r74263 * r74263;
        double r74265 = r74262 * r74264;
        double r74266 = a2;
        double r74267 = r74266 * r74266;
        double r74268 = r74262 * r74267;
        double r74269 = r74265 + r74268;
        return r74269;
}
↓
double f(double a1, double a2, double th) {
        double r74270 = th;
        double r74271 = cos(r74270);
        double r74272 = a2;
        double r74273 = r74272 * r74272;
        double r74274 = 2.0;
        double r74275 = cbrt(r74274);
        double r74276 = sqrt(r74275);
        double r74277 = sqrt(r74276);
        double r74278 = sqrt(r74274);
        double r74279 = sqrt(r74278);
        double r74280 = r74277 * r74279;
        double r74281 = r74273 / r74280;
        double r74282 = r74275 * r74275;
        double r74283 = sqrt(r74282);
        double r74284 = sqrt(r74283);
        double r74285 = r74281 / r74284;
        double r74286 = a1;
        double r74287 = r74286 * r74286;
        double r74288 = r74287 / r74278;
        double r74289 = r74285 + r74288;
        double r74290 = r74271 * r74289;
        return r74290;
}

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

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied add-cube-cbrt0.6

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

  8. Applied sqrt-prod0.6

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

  9. Applied sqrt-prod0.6

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

  10. Applied div-inv0.5

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

  11. Applied times-frac0.5

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

  12. Applied associate-*l*0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019298 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))