Average Error: 1.0 → 0.0
Time: 31.8s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\sqrt{\pi} \cdot \left(\frac{2}{3} \cdot \sqrt{\pi}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\sqrt{\pi} \cdot \left(\frac{2}{3} \cdot \sqrt{\pi}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2
double f(double g, double h) {
        double r6467147 = 2.0;
        double r6467148 = atan2(1.0, 0.0);
        double r6467149 = r6467147 * r6467148;
        double r6467150 = 3.0;
        double r6467151 = r6467149 / r6467150;
        double r6467152 = g;
        double r6467153 = -r6467152;
        double r6467154 = h;
        double r6467155 = r6467153 / r6467154;
        double r6467156 = acos(r6467155);
        double r6467157 = r6467156 / r6467150;
        double r6467158 = r6467151 + r6467157;
        double r6467159 = cos(r6467158);
        double r6467160 = r6467147 * r6467159;
        return r6467160;
}
↓
double f(double g, double h) {
        double r6467161 = g;
        double r6467162 = -r6467161;
        double r6467163 = h;
        double r6467164 = r6467162 / r6467163;
        double r6467165 = acos(r6467164);
        double r6467166 = 3.0;
        double r6467167 = r6467165 / r6467166;
        double r6467168 = cos(r6467167);
        double r6467169 = 2.0;
        double r6467170 = r6467169 / r6467166;
        double r6467171 = atan2(1.0, 0.0);
        double r6467172 = r6467170 * r6467171;
        double r6467173 = cos(r6467172);
        double r6467174 = r6467168 * r6467173;
        double r6467175 = sqrt(r6467171);
        double r6467176 = r6467170 * r6467175;
        double r6467177 = r6467175 * r6467176;
        double r6467178 = sin(r6467177);
        double r6467179 = sin(r6467167);
        double r6467180 = r6467178 * r6467179;
        double r6467181 = r6467174 - r6467180;
        double r6467182 = r6467181 * r6467169;
        return r6467182;
}

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

  2. Simplified1.0

    \[\leadsto \color{blue}{2 \cdot \cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\]
  3. Using strategy rm

  4. Applied fma-udef1.0

    \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{2}{3} \cdot \pi + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)}\]

  5. Applied cos-sum1.0

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) - \sin \left(\frac{2}{3} \cdot \color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\]

  8. Applied associate-*r*0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) - \sin \color{blue}{\left(\left(\frac{2}{3} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\]

  9. Final simplification0.0

    \[\leadsto \left(\cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\sqrt{\pi} \cdot \left(\frac{2}{3} \cdot \sqrt{\pi}\right)\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))