Average Error: 1.0 → 0.0
Time: 4.3s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right) - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \pi}{3}\right)\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right) - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \pi}{3}\right)\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right)
double f(double g, double h) {
        double r137383 = 2.0;
        double r137384 = atan2(1.0, 0.0);
        double r137385 = r137383 * r137384;
        double r137386 = 3.0;
        double r137387 = r137385 / r137386;
        double r137388 = g;
        double r137389 = -r137388;
        double r137390 = h;
        double r137391 = r137389 / r137390;
        double r137392 = acos(r137391);
        double r137393 = r137392 / r137386;
        double r137394 = r137387 + r137393;
        double r137395 = cos(r137394);
        double r137396 = r137383 * r137395;
        return r137396;
}
↓
double f(double g, double h) {
        double r137397 = 2.0;
        double r137398 = atan2(1.0, 0.0);
        double r137399 = r137397 * r137398;
        double r137400 = 3.0;
        double r137401 = r137399 / r137400;
        double r137402 = cos(r137401);
        double r137403 = g;
        double r137404 = -r137403;
        double r137405 = h;
        double r137406 = r137404 / r137405;
        double r137407 = acos(r137406);
        double r137408 = sqrt(r137407);
        double r137409 = sqrt(r137400);
        double r137410 = r137408 / r137409;
        double r137411 = r137410 * r137410;
        double r137412 = cos(r137411);
        double r137413 = r137402 * r137412;
        double r137414 = log1p(r137401);
        double r137415 = expm1(r137414);
        double r137416 = sin(r137415);
        double r137417 = sin(r137411);
        double r137418 = r137416 * r137417;
        double r137419 = r137413 - r137418;
        double r137420 = r137397 * r137419;
        return r137420;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt1.0

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied times-frac1.0

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

  7. Applied cos-sum1.0

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

  9. Applied expm1-log1p-u0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right) - \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \pi}{3}\right)\right)\right)} \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right)\]

  10. Final simplification0.0

    \[\leadsto 2 \cdot \left(\cos \left(\frac{2 \cdot \pi}{3}\right) \cdot \cos \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right) - \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \pi}{3}\right)\right)\right) \cdot \sin \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}} \cdot \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\sqrt{3}}\right)\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))