2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 10.2s

Alternatives: 5

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.6666666666666666\\ t_1 := \cos^{-1} \left(\frac{-g}{h}\right)\\ \left(-2\right) \cdot \frac{{\left(-\cos \left(\mathsf{fma}\left(t\_1, 0.3333333333333333, t\_0\right)\right)\right)}^{3}}{\cos \left(\mathsf{fma}\left(0.3333333333333333, t\_1, t\_0\right) \cdot 2\right) \cdot 0.5 + 0.5} \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* (PI) 0.6666666666666666)) (t_1 (acos (/ (- g) h))))
   (*
    (- 2.0)
    (/
     (pow (- (cos (fma t_1 0.3333333333333333 t_0))) 3.0)
     (+ (* (cos (* (fma 0.3333333333333333 t_1 t_0) 2.0)) 0.5) 0.5)))))

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%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

6. Applied rewrites 99.9%

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

8. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{0 + \left({\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + 0 \cdot \left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{0 + \left({\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + 0 \cdot \left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

   2. +-lft-identity 99.9

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + 0 \cdot \left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + 0 \cdot \left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{0 \cdot \left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

   5. mul0-lft N/A

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + \color{blue}{0}} \]

   6. +-commutative N/A

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{0 + {\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}} \]

   7. +-lft-identity 99.9

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}} \]

   8. lift-pow.f64 N/A

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}} \]

   9. unpow2 N/A

      \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

   10. lift-cos.f64 N/A

       \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   11. lift-cos.f64 N/A

       \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

   12. sqr-cos-a N/A

       \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1}{3}, \frac{2}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

10. Applied rewrites 99.9%

    \[\leadsto 2 \cdot \frac{0 - {\left(-\cos \left(\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 0.3333333333333333, 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{3}}{\color{blue}{0.5 + 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), 0.6666666666666666 \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

11. Final simplification 99.9%

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

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1.5}{\mathsf{PI}\left(\right)}\\ \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), t\_0, 3\right)}{t\_0 \cdot 3}\right) \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (/ 1.5 (PI))))
   (* (cos (/ (fma (acos (/ (- g) h)) t_0 3.0) (* t_0 3.0))) 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. clear-num N/A

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

   6. frac-add N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/r* N/A

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

   17. lower-/.f64 N/A

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

   18. metadata-eval 98.5

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1.5}{\mathsf{PI}\left(\right)}, 3\right)}{3 \cdot \frac{\color{blue}{1.5}}{\mathsf{PI}\left(\right)}}\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1.5}{\mathsf{PI}\left(\right)}, 3\right)}{3 \cdot \frac{1.5}{\mathsf{PI}\left(\right)}}\right)} \]

5. Final simplification 98.5%

   \[\leadsto \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), \frac{1.5}{\mathsf{PI}\left(\right)}, 3\right)}{\frac{1.5}{\mathsf{PI}\left(\right)} \cdot 3}\right) \cdot 2 \]
6. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{-g}{h}\right), 3, 6 \cdot \mathsf{PI}\left(\right)\right)}{9}\right) \cdot 2 \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (/ (fma (acos (/ (- g) h)) 3.0 (* 6.0 (PI))) 9.0)) 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. lift-/.f64 N/A

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

   4. frac-add N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (*
  (cos (fma (PI) 0.6666666666666666 (* 0.3333333333333333 (acos (/ (- g) h)))))
  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 98.4

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

   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 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 5: 98.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (* (fma (PI) 2.0 (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. 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

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

   5. div-inv N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))