Average Error: 0.5 → 0.5
Time: 1.2m
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(a2 \cdot a2\right) \cdot \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a1 \cdot a1\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(a2 \cdot a2\right) \cdot \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} + \frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a1 \cdot a1\right)
double f(double a1, double a2, double th) {
        double r3115100 = th;
        double r3115101 = cos(r3115100);
        double r3115102 = 2.0;
        double r3115103 = sqrt(r3115102);
        double r3115104 = r3115101 / r3115103;
        double r3115105 = a1;
        double r3115106 = r3115105 * r3115105;
        double r3115107 = r3115104 * r3115106;
        double r3115108 = a2;
        double r3115109 = r3115108 * r3115108;
        double r3115110 = r3115104 * r3115109;
        double r3115111 = r3115107 + r3115110;
        return r3115111;
}
↓
double f(double a1, double a2, double th) {
        double r3115112 = a2;
        double r3115113 = r3115112 * r3115112;
        double r3115114 = th;
        double r3115115 = cos(r3115114);
        double r3115116 = 2.0;
        double r3115117 = sqrt(r3115116);
        double r3115118 = sqrt(r3115117);
        double r3115119 = r3115115 / r3115118;
        double r3115120 = r3115119 / r3115118;
        double r3115121 = r3115113 * r3115120;
        double r3115122 = a1;
        double r3115123 = r3115122 * r3115122;
        double r3115124 = r3115120 * r3115123;
        double r3115125 = r3115121 + r3115124;
        return r3115125;
}

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-sqr-sqrt0.5

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

  8. Applied associate-/r*0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019139 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))