2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 3.8s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[2 \cdot \sin \left(\mathsf{fma}\left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (sin
   (fma
    (cbrt (* (PI) (PI)))
    (* (cbrt (PI)) 0.5)
    (fma
     -0.3333333333333333
     (acos (/ (- g) h))
     (* -0.6666666666666666 (PI)))))))

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

3. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \mathsf{PI}\left(\right) \cdot 0.5\right)} \]
4. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot 0.5, \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\sin \left(\mathsf{fma}\left(-0.6666666666666666, \mathsf{PI}\left(\right), \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 1.5707963267948966\right)\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (sin
   (fma
    -0.6666666666666666
    (PI)
    (fma -0.3333333333333333 (acos (/ (- g) h)) 1.5707963267948966)))
  2.0))

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

3. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \mathsf{PI}\left(\right) \cdot 0.5\right)} \]

4. Evaluated real constant 98.5%

   \[\leadsto 2 \cdot \sin \left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \color{blue}{\frac{884279719003555}{562949953421312}}\right) \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3} + \frac{884279719003555}{562949953421312}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{\mathsf{fma}\left(\mathsf{PI}\left(\right), 2, \cos^{-1} \left(\frac{-g}{h}\right)\right)}{-3}} + \frac{884279719003555}{562949953421312}\right) \]

   3. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \sin \left(\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot 2 + \cos^{-1} \left(\frac{-g}{h}\right)}}{-3} + \frac{884279719003555}{562949953421312}\right) \]

   4. div-add N/A

      \[\leadsto 2 \cdot \sin \left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{-3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{-3}\right)} + \frac{884279719003555}{562949953421312}\right) \]

   5. associate-/l* N/A

      \[\leadsto 2 \cdot \sin \left(\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{2}{-3}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{-3}\right) + \frac{884279719003555}{562949953421312}\right) \]

   6. metadata-eval N/A

      \[\leadsto 2 \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-2}{3}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{-3}\right) + \frac{884279719003555}{562949953421312}\right) \]

   7. *-commutative N/A

      \[\leadsto 2 \cdot \sin \left(\left(\color{blue}{\frac{-2}{3} \cdot \mathsf{PI}\left(\right)} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{-3}\right) + \frac{884279719003555}{562949953421312}\right) \]

   8. mult-flip-rev N/A

      \[\leadsto 2 \cdot \sin \left(\left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right) + \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{1}{-3}}\right) + \frac{884279719003555}{562949953421312}\right) \]

   9. metadata-eval N/A

      \[\leadsto 2 \cdot \sin \left(\left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(\frac{-g}{h}\right) \cdot \color{blue}{\frac{-1}{3}}\right) + \frac{884279719003555}{562949953421312}\right) \]

   10. *-commutative N/A

       \[\leadsto 2 \cdot \sin \left(\left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)}\right) + \frac{884279719003555}{562949953421312}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\left(\color{blue}{\frac{-2}{3} \cdot \mathsf{PI}\left(\right)} + \frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) + \frac{884279719003555}{562949953421312}\right) \]

   12. associate-+l+ N/A

       \[\leadsto 2 \cdot \sin \color{blue}{\left(\frac{-2}{3} \cdot \mathsf{PI}\left(\right) + \left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{884279719003555}{562949953421312}\right)\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto 2 \cdot \sin \left(\color{blue}{\frac{-2}{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{884279719003555}{562949953421312}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto 2 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{-2}{3}} + \left(\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{884279719003555}{562949953421312}\right)\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{-2}{3}, \frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right) + \frac{884279719003555}{562949953421312}\right)\right)} \]

   16. *-commutative N/A

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{-2}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3}} + \frac{884279719003555}{562949953421312}\right)\right) \]

   17. lower-fma.f64 99.9

       \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.6666666666666666, \color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, 1.5707963267948966\right)}\right)\right) \]

6. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.6666666666666666, \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, 1.5707963267948966\right)\right)\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{-2}{3}, \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \frac{884279719003555}{562949953421312}\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{-2}{3}, \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \frac{884279719003555}{562949953421312}\right)\right)\right) \cdot 2} \]

   3. lower-*.f64 99.9

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.6666666666666666, \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, 1.5707963267948966\right)\right)\right) \cdot 2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{-2}{3} + \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \frac{884279719003555}{562949953421312}\right)\right)} \cdot 2 \]

   5. *-commutative N/A

      \[\leadsto \sin \left(\color{blue}{\frac{-2}{3} \cdot \mathsf{PI}\left(\right)} + \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{-1}{3}, \frac{884279719003555}{562949953421312}\right)\right) \cdot 2 \]

   6. lower-fma.f64 99.9

      \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(-0.6666666666666666, \mathsf{PI}\left(\right), \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -0.3333333333333333, 1.5707963267948966\right)\right)\right)} \cdot 2 \]

   7. lift-fma.f64 N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\frac{-2}{3}, \mathsf{PI}\left(\right), \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \frac{-1}{3} + \frac{884279719003555}{562949953421312}}\right)\right) \cdot 2 \]

   8. *-commutative N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\frac{-2}{3}, \mathsf{PI}\left(\right), \color{blue}{\frac{-1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)} + \frac{884279719003555}{562949953421312}\right)\right) \cdot 2 \]

   9. lower-fma.f64 99.9

      \[\leadsto \sin \left(\mathsf{fma}\left(-0.6666666666666666, \mathsf{PI}\left(\right), \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 1.5707963267948966\right)}\right)\right) \cdot 2 \]

8. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(-0.6666666666666666, \mathsf{PI}\left(\right), \mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 1.5707963267948966\right)\right)\right) \cdot 2} \]
9. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\cos \left(\mathsf{fma}\left(-0.3333333333333333, \mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right), -2.0943951023931957\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (fma -0.3333333333333333 (- (PI) (acos (/ g h))) -2.0943951023931957))
  2.0))

Derivation

1. Initial program 98.5%

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

2. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{2358079250676147}{1125899906842624}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

   3. lower-*.f64 98.4

      \[\leadsto \color{blue}{\cos \left(2.0943951023931957 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2} \]
5. Step-by-step derivation

   1. lift-acos.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\cos^{-1} \left(\frac{-g}{h}\right)}, \frac{-2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   2. lift-/.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \color{blue}{\left(\frac{-g}{h}\right)}, \frac{-2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   3. lift-neg.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right), \frac{-2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   4. distribute-frac-neg N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{-1}{3}, \cos^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{g}{h}\right)\right)}, \frac{-2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   5. acos-neg N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}, \frac{-2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   6. lift-PI.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right)} - \cos^{-1} \left(\frac{g}{h}\right), \frac{-2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   7. lower--.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}, \frac{-2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   8. lower-acos.f64 N/A

      \[\leadsto \cos \left(\mathsf{fma}\left(\frac{-1}{3}, \mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left(\frac{g}{h}\right)}, \frac{-2358079250676147}{1125899906842624}\right)\right) \cdot 2 \]

   9. lower-/.f64 98.5

      \[\leadsto \cos \left(\mathsf{fma}\left(-0.3333333333333333, \mathsf{PI}\left(\right) - \cos^{-1} \color{blue}{\left(\frac{g}{h}\right)}, -2.0943951023931957\right)\right) \cdot 2 \]

6. Applied rewrites 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(-0.3333333333333333, \color{blue}{\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)}, -2.0943951023931957\right)\right) \cdot 2 \]
7. Add Preprocessing

Alternative 4: 98.4% accurate, 1.2× speedup

Math

\[\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (fma -0.3333333333333333 (acos (/ (- g) h)) -2.0943951023931957))
  2.0))

C

double code(double g, double h) {
	return cos(fma(-0.3333333333333333, acos((-g / h)), -2.0943951023931957)) * 2.0;
}

Julia

function code(g, h)
	return Float64(cos(fma(-0.3333333333333333, acos(Float64(Float64(-g) / h)), -2.0943951023931957)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Cos[N[(-0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + -2.0943951023931957), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

2. Evaluated real constant 98.4%

   \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{2358079250676147}{1125899906842624}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\frac{2358079250676147}{1125899906842624} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

   3. lower-*.f64 98.4

      \[\leadsto \color{blue}{\cos \left(2.0943951023931957 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot 2} \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(-0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), -2.0943951023931957\right)\right) \cdot 2} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.2× speedup

Math

\[\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 3.6651914291880923\right)\right) \cdot 2 \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (sin (fma 0.3333333333333333 (acos (/ (- g) h)) 3.6651914291880923)) 2.0))

C

double code(double g, double h) {
	return sin(fma(0.3333333333333333, acos((-g / h)), 3.6651914291880923)) * 2.0;
}

Julia

function code(g, h)
	return Float64(sin(fma(0.3333333333333333, acos(Float64(Float64(-g) / h)), 3.6651914291880923)) * 2.0)
end

Wolfram

code[g_, h_] := N[(N[Sin[N[(0.3333333333333333 * N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] + 3.6651914291880923), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

3. Applied rewrites 97.5%

   \[\leadsto 2 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \mathsf{fma}\left(\mathsf{PI}\left(\right), 0.6666666666666666, \mathsf{PI}\left(\right) \cdot 0.5\right)\right)\right)} \]

4. Evaluated real constant 97.6%

   \[\leadsto 2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \color{blue}{\frac{4126638688683257}{1125899906842624}}\right)\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{4126638688683257}{1125899906842624}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\frac{1}{3}, \cos^{-1} \left(\frac{-g}{h}\right), \frac{4126638688683257}{1125899906842624}\right)\right) \cdot 2} \]

   3. lower-*.f64 97.6

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 3.6651914291880923\right)\right) \cdot 2} \]

6. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 3.6651914291880923\right)\right) \cdot 2} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))