Average Error: 0.5 → 0.4
Time: 12.7s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \left(e^{\cos x - \cos y}\right)}{3 \cdot \left(\left(1 + \sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
↓
\frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \left(e^{\cos x - \cos y}\right)}{3 \cdot \left(\left(1 + \sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}
double f(double x, double y) {
        double r211582 = 2.0;
        double r211583 = sqrt(r211582);
        double r211584 = x;
        double r211585 = sin(r211584);
        double r211586 = y;
        double r211587 = sin(r211586);
        double r211588 = 16.0;
        double r211589 = r211587 / r211588;
        double r211590 = r211585 - r211589;
        double r211591 = r211583 * r211590;
        double r211592 = r211585 / r211588;
        double r211593 = r211587 - r211592;
        double r211594 = r211591 * r211593;
        double r211595 = cos(r211584);
        double r211596 = cos(r211586);
        double r211597 = r211595 - r211596;
        double r211598 = r211594 * r211597;
        double r211599 = r211582 + r211598;
        double r211600 = 3.0;
        double r211601 = 1.0;
        double r211602 = 5.0;
        double r211603 = sqrt(r211602);
        double r211604 = r211603 - r211601;
        double r211605 = r211604 / r211582;
        double r211606 = r211605 * r211595;
        double r211607 = r211601 + r211606;
        double r211608 = r211600 - r211603;
        double r211609 = r211608 / r211582;
        double r211610 = r211609 * r211596;
        double r211611 = r211607 + r211610;
        double r211612 = r211600 * r211611;
        double r211613 = r211599 / r211612;
        return r211613;
}
↓
double f(double x, double y) {
        double r211614 = 2.0;
        double r211615 = sqrt(r211614);
        double r211616 = sqrt(r211615);
        double r211617 = x;
        double r211618 = sin(r211617);
        double r211619 = y;
        double r211620 = sin(r211619);
        double r211621 = 16.0;
        double r211622 = r211620 / r211621;
        double r211623 = r211618 - r211622;
        double r211624 = r211616 * r211623;
        double r211625 = r211616 * r211624;
        double r211626 = r211618 / r211621;
        double r211627 = r211620 - r211626;
        double r211628 = r211625 * r211627;
        double r211629 = cos(r211617);
        double r211630 = cos(r211619);
        double r211631 = r211629 - r211630;
        double r211632 = exp(r211631);
        double r211633 = log(r211632);
        double r211634 = r211628 * r211633;
        double r211635 = r211614 + r211634;
        double r211636 = 3.0;
        double r211637 = 1.0;
        double r211638 = 5.0;
        double r211639 = sqrt(r211638);
        double r211640 = r211639 - r211637;
        double r211641 = r211640 / r211614;
        double r211642 = sqrt(r211641);
        double r211643 = r211642 * r211629;
        double r211644 = r211642 * r211643;
        double r211645 = r211637 + r211644;
        double r211646 = r211636 * r211636;
        double r211647 = -r211638;
        double r211648 = r211646 + r211647;
        double r211649 = r211636 + r211639;
        double r211650 = r211648 / r211649;
        double r211651 = r211650 / r211614;
        double r211652 = r211651 * r211630;
        double r211653 = r211645 + r211652;
        double r211654 = r211636 * r211653;
        double r211655 = r211635 / r211654;
        return r211655;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  4. Applied sqrt-prod0.5

    \[\leadsto \frac{2 + \left(\left(\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  5. Applied associate-*l*0.5

    \[\leadsto \frac{2 + \left(\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
  6. Using strategy rm

  7. Applied flip--0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2} \cdot \cos y\right)}\]

  8. Simplified0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{\color{blue}{3 \cdot 3 + \left(-5\right)}}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \sqrt{\frac{\sqrt{5} - 1}{2}}\right)} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  11. Applied associate-*l*0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \color{blue}{\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)}\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]
  12. Using strategy rm

  13. Applied add-log-exp0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \color{blue}{\log \left(e^{\cos y}\right)}\right)}{3 \cdot \left(\left(1 + \sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  14. Applied add-log-exp0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\color{blue}{\log \left(e^{\cos x}\right)} - \log \left(e^{\cos y}\right)\right)}{3 \cdot \left(\left(1 + \sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  15. Applied diff-log0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \color{blue}{\log \left(\frac{e^{\cos x}}{e^{\cos y}}\right)}}{3 \cdot \left(\left(1 + \sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  16. Simplified0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \color{blue}{\left(e^{\cos x - \cos y}\right)}}{3 \cdot \left(\left(1 + \sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

  17. Final simplification0.4

    \[\leadsto \frac{2 + \left(\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \log \left(e^{\cos x - \cos y}\right)}{3 \cdot \left(\left(1 + \sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \left(\sqrt{\frac{\sqrt{5} - 1}{2}} \cdot \cos x\right)\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y\right)}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2 (* (* (* (sqrt 2) (- (sin x) (/ (sin y) 16))) (- (sin y) (/ (sin x) 16))) (- (cos x) (cos y)))) (* 3 (+ (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x))) (* (/ (- 3 (sqrt 5)) 2) (cos y))))))