2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 4.2s

Alternatives: 6

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, 0.3× speedup

Math

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

FPCore

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

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. Add Preprocessing

4. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (cos (/ (fma (PI) 2.0 (acos (/ (- g) h))) -3.0))))
   (* (pow t_0 3.0) (/ 2.0 (pow t_0 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. flip3-+ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. count-2 N/A

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

   3. lower-*.f64 98.5

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

6. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. +-lft-identity N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

8. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(\frac{-g}{h}\right)\\ \frac{{\cos \left(\mathsf{fma}\left(0.3333333333333333, t\_0, \mathsf{PI}\left(\right) \cdot 0.6666666666666666\right)\right)}^{3}}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(0.6666666666666666, t\_0, 1.3333333333333333 \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right)} \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (acos (/ (- g) h))))
   (*
    (/
     (pow (cos (fma 0.3333333333333333 t_0 (* (PI) 0.6666666666666666))) 3.0)
     (fma
      (cos (fma 0.6666666666666666 t_0 (* 1.3333333333333333 (PI))))
      0.5
      0.5))
    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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. flip3-+ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. count-2 N/A

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

   3. lower-*.f64 98.5

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

6. Applied rewrites 98.5%

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

   1. lift-+.f64 N/A

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

   2. +-lft-identity 98.5

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-acos.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-/.f64 N/A

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

8. Applied rewrites 97.6%

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

9. Taylor expanded in g around 0

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

11. Applied rewrites 99.9%

    \[\leadsto \color{blue}{\frac{{\cos \left(\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\frac{-g}{h}\right), \mathsf{PI}\left(\right) \cdot 0.6666666666666666\right)\right)}^{3}}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(0.6666666666666666, \cos^{-1} \left(\frac{-g}{h}\right), 1.3333333333333333 \cdot \mathsf{PI}\left(\right)\right)\right), 0.5, 0.5\right)} \cdot 2} \]
12. Add Preprocessing

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (* (cos (/ (fma (PI) 2.0 (acos (/ (- g) h))) 3.0)) 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.5

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

4. Applied rewrites 98.5%

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

Alternative 5: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

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. Add Preprocessing

3. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lower-acos.f64 N/A

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

   9. distribute-frac-neg N/A

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

   10. mul-1-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. mul-1-neg N/A

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

   14. distribute-frac-neg N/A

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

   15. lift-/.f64 N/A

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

   16. lift-neg.f64 N/A

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

   17. lift-acos.f64 N/A

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

   18. lower-*.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 6: 98.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos
   (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. Add Preprocessing

3. Taylor expanded in g around 0

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-acos.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 98.5

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

5. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2025038 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 (PI)) 3.0) (/ (acos (/ (- g) h)) 3.0)))))