Average Error: 1.0 → 0.0
Time: 20.9s
Precision: binary64
Cost: 2560
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{0.1111111111111111 \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{0.1111111111111111 \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)
(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))
↓
(FPCore (g h)
 :precision binary64
 (*
  2.0
  (+
   (*
    (cos (+ (* PI 0.6666666666666666) (/ PI 3.0)))
    (cos (/ (acos (/ g h)) 3.0)))
   (*
    (sin
     (+
      (* PI 0.6666666666666666)
      (cbrt (* 0.1111111111111111 (* (pow PI 3.0) 0.3333333333333333)))))
    (sin (/ (acos (/ g h)) 3.0))))))
double code(double g, double h) {
	return 2.0 * cos(((2.0 * ((double) M_PI)) / 3.0) + (acos(-g / h) / 3.0));
}
↓
double code(double g, double h) {
	return 2.0 * ((cos((((double) M_PI) * 0.6666666666666666) + (((double) M_PI) / 3.0)) * cos(acos(g / h) / 3.0)) + (sin((((double) M_PI) * 0.6666666666666666) + cbrt(0.1111111111111111 * (pow(((double) M_PI), 3.0) * 0.3333333333333333))) * sin(acos(g / h) / 3.0)));
}

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Accuracy1.0
Cost1536
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{\frac{3}{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}}\right)\]

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 distribute-frac-neg_binary64_34511.0

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

  4. Applied acos-neg_binary64_36711.0

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

  5. Applied div-sub_binary64_34931.0

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

  6. Applied associate-+r-_binary64_34221.0

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

  7. Applied cos-diff_binary64_36250.0

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

  8. Simplified0.0

    \[\leadsto 2 \cdot \left(\color{blue}{\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)} + \sin \left(\frac{2 \cdot \pi}{3} + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  9. Simplified0.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \color{blue}{\sin \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)\]
  10. Using strategy rm

  11. Applied add-cbrt-cube_binary64_35240.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \color{blue}{\sqrt[3]{\left(\frac{\pi}{3} \cdot \frac{\pi}{3}\right) \cdot \frac{\pi}{3}}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  12. Simplified0.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{\color{blue}{{\left(\frac{\pi}{3}\right)}^{3}}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
  13. Using strategy rm

  14. Applied add-cube-cbrt_binary64_35230.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{{\left(\frac{\pi}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}\right)}^{3}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  15. Applied *-un-lft-identity_binary64_34880.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{{\left(\frac{\color{blue}{1 \cdot \pi}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}\right)}^{3}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  16. Applied times-frac_binary64_34940.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{{\color{blue}{\left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\pi}{\sqrt[3]{3}}\right)}}^{3}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
  17. Using strategy rm

  18. Applied unpow-prod-down_binary64_35670.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{\color{blue}{{\left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)}^{3} \cdot {\left(\frac{\pi}{\sqrt[3]{3}}\right)}^{3}}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  19. Simplified0.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{\color{blue}{0.1111111111111111} \cdot {\left(\frac{\pi}{\sqrt[3]{3}}\right)}^{3}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  20. Simplified0.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{0.1111111111111111 \cdot \color{blue}{\left({\pi}^{3} \cdot 0.3333333333333333\right)}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  21. Final simplification0.0

    \[\leadsto 2 \cdot \left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \sqrt[3]{0.1111111111111111 \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

Reproduce

herbie shell --seed 2020338 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))