2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 8.8s

Alternatives: 3

Speedup: 1.1×

Specification

Math

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

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

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

FPCore

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ t_1 := \cos \left(\mathsf{fma}\left(0.3333333333333333, t\_0, -0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\\ \frac{2 \cdot \cos \left(\mathsf{fma}\left(0.3333333333333333, t\_0, \mathsf{PI}\left(\right) \cdot 0.6666666666666666\right)\right)}{t\_1} \cdot t\_1 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h)))
        (t_1 (cos (fma 0.3333333333333333 t_0 (* -0.6666666666666666 (PI))))))
   (*
    (/
     (* 2.0 (cos (fma 0.3333333333333333 t_0 (* (PI) 0.6666666666666666))))
     t_1)
    t_1)))

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.4%

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos \left(\mathsf{fma}\left(-0.1111111111111111 \cdot \mathsf{PI}\left(\right), -6, \cos^{-1} \left(\frac{-g}{h}\right) \cdot 0.3333333333333333\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos
   (fma
    (* -0.1111111111111111 (PI))
    -6.0
    (* (acos (/ (- g) h)) 0.3333333333333333)))
  2.0))

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. frac-add N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{\color{blue}{-9}}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + 3 \cdot \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \]

   12. distribute-rgt-neg-out N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \color{blue}{\left(\mathsf{neg}\left(3 \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}}\right) \]

   13. *-commutative N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot 3}\right)\right)}}\right) \]

   14. distribute-lft-neg-out N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\cos^{-1} \left(\frac{-g}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3}}}\right) \]

   15. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{\frac{-9}{\color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \mathsf{neg}\left(3\right), \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 3\right)}}}\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}\right)} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{\color{blue}{\frac{-9}{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)}}}\right) \]

   3. associate-/r/ N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{-9} \cdot \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), -3, -6 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{-9} \cdot \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3 + -6 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   5. +-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\frac{1}{-9} \cdot \color{blue}{\left(-6 \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right)}\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(-6 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{-9} + \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(-6 \cdot \mathsf{PI}\left(\right), \frac{1}{-9}, \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{-6 \cdot \mathsf{PI}\left(\right)}, \frac{1}{-9}, \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot -6}, \frac{1}{-9}, \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot -6}, \frac{1}{-9}, \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \color{blue}{\frac{-1}{9}}, \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \frac{-1}{9}, \color{blue}{\left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot -3\right) \cdot \frac{1}{-9}}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \frac{-1}{9}, \color{blue}{\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot \frac{1}{-9}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, \frac{-1}{9}, \color{blue}{\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)} \cdot \frac{1}{-9}\right)\right) \]

   15. metadata-eval 98.5

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, -0.1111111111111111, \left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \color{blue}{-0.1111111111111111}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot -6, -0.1111111111111111, \left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot -0.1111111111111111\right)\right)} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot -6\right) \cdot \frac{-1}{9} + \left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right)} \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{-1}{9} \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} + \left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{-1}{9} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot -6\right)} + \left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \]

   4. associate-*r* N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot -6} + \left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot -6 + \color{blue}{\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \cdot \frac{-1}{9}}\right) \]

   6. *-commutative N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot -6 + \color{blue}{\frac{-1}{9} \cdot \left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot -6 + \frac{-1}{9} \cdot \color{blue}{\left(-3 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right) \]

   8. associate-*r* N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot -6 + \color{blue}{\left(\frac{-1}{9} \cdot -3\right) \cdot \cos^{-1} \left(\frac{-g}{h}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto 2 \cdot \cos \left(\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right)\right) \cdot -6 + \color{blue}{\frac{1}{3}} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{9} \cdot \mathsf{PI}\left(\right), -6, \frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{9} \cdot \mathsf{PI}\left(\right)}, -6, \frac{1}{3} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right) \]

   12. lower-*.f64 98.5

       \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(-0.1111111111111111 \cdot \mathsf{PI}\left(\right), -6, \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)}\right)\right) \]

8. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(-0.1111111111111111 \cdot \mathsf{PI}\left(\right), -6, 0.3333333333333333 \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)\right)} \]

9. Final simplification 98.5%

   \[\leadsto \cos \left(\mathsf{fma}\left(-0.1111111111111111 \cdot \mathsf{PI}\left(\right), -6, \cos^{-1} \left(\frac{-g}{h}\right) \cdot 0.3333333333333333\right)\right) \cdot 2 \]
10. Add Preprocessing

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval 98.5

      \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \color{blue}{0.6666666666666666}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \]

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))