UniformSampleCone, x

Percentage Accurate: 57.5% → 99.0%

Time: 8.1s

Alternatives: 13

Speedup: 5.8×

Specification

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0))))))

Initial Program: 57.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0))))))

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(2 - {\left(maxCos - 1\right)}^{2} \cdot ux, ux, \left(-2 \cdot maxCos\right) \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) (PI)))
  (sqrt
   (fma (- 2.0 (* (pow (- maxCos 1.0) 2.0) ux)) ux (* (* -2.0 maxCos) ux)))))

Derivation

1. Initial program 54.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux}} \]
6. Step-by-step derivation

7. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(2 - {\left(maxCos - 1\right)}^{2} \cdot ux, \color{blue}{ux}, \left(-2 \cdot maxCos\right) \cdot ux\right)} \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) (PI)))
  (sqrt (* (fma -2.0 maxCos (- 2.0 (* (pow (- maxCos 1.0) 2.0) ux))) ux))))

Derivation

1. Initial program 54.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux}} \]
6. Add Preprocessing

Alternative 3: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.004000000189989805:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\left(\mathsf{fma}\left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right), ux, \left(-maxCos\right) - -1\right) - -1\right) - maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<=
        (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0))))
        0.004000000189989805)
     (*
      (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0)
      (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (*
      1.0
      (sqrt
       (*
        (-
         (-
          (fma (* (- 1.0 maxCos) (- maxCos 1.0)) ux (- (- maxCos) -1.0))
          -1.0)
         maxCos)
        ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.00400000019

   1. Initial program 29.2%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.6%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]

   6. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]

   if 0.00400000019 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 82.2%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 72.4%

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around inf

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(maxCos \cdot \left(\left(ux + \frac{1}{maxCos}\right) - \frac{ux}{maxCos}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.3%

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]
   9. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right) + 1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)}} \]

      6. lower-neg.f32 71.3

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      8. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      9. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      10. lower-fma.f32 71.3

          \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

   10. Applied rewrites 71.3%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \left(\frac{1 - ux}{maxCos} + ux\right) \cdot maxCos, 1\right)}} \]

   11. Taylor expanded in ux around 0

       \[\leadsto 1 \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 84.7%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right), ux, -\left(maxCos - 1\right)\right) + 1\right) - maxCos\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.004000000189989805:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\left(\mathsf{fma}\left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right), ux, \left(-maxCos\right) - -1\right) - -1\right) - maxCos\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.02199999988079071:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \mathsf{fma}\left(maxCos, ux, 1 - ux\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<=
        (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0))))
        0.02199999988079071)
     (*
      (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0)
      (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (sqrt (fma (- ux (fma maxCos ux 1.0)) (fma maxCos ux (- 1.0 ux)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0219999999

   1. Initial program 37.3%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

   5. Applied rewrites 90.9%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]

   6. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]

   if 0.0219999999 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 89.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around inf

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(maxCos \cdot \left(\left(ux + \frac{1}{maxCos}\right) - \frac{ux}{maxCos}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.2%

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]
   9. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right) + 1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)}} \]

      6. lower-neg.f32 77.2

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      8. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      9. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      10. lower-fma.f32 77.2

          \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

   10. Applied rewrites 77.2%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \left(\frac{1 - ux}{maxCos} + ux\right) \cdot maxCos, 1\right)}} \]

   11. Taylor expanded in uy around 0

       \[\leadsto \color{blue}{\sqrt{1 + \left(ux - \left(1 + maxCos \cdot ux\right)\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 78.2%

       \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \mathsf{fma}\left(maxCos, ux, 1 - ux\right), 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.025200000032782555:\\ \;\;\;\;1 \cdot \sqrt{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\mathsf{fma}\left(ux - 1, 1 - ux, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<=
        (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0))))
        0.025200000032782555)
     (* 1.0 (sqrt (* (- (fma -1.0 (- maxCos 1.0) 1.0) maxCos) ux)))
     (* 1.0 (sqrt (fma (- ux 1.0) (- 1.0 ux) 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0252

   1. Initial program 37.7%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.3%

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around inf

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(maxCos \cdot \left(\left(ux + \frac{1}{maxCos}\right) - \frac{ux}{maxCos}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.9%

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]
   9. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right) + 1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)}} \]

      6. lower-neg.f32 30.9

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      8. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      9. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      10. lower-fma.f32 30.9

          \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

   10. Applied rewrites 30.9%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \left(\frac{1 - ux}{maxCos} + ux\right) \cdot maxCos, 1\right)}} \]

   11. Taylor expanded in ux around 0

       \[\leadsto 1 \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 69.6%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}} \]

   if 0.0252 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 89.4%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around inf

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(maxCos \cdot \left(\left(ux + \frac{1}{maxCos}\right) - \frac{ux}{maxCos}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.3%

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]
   9. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right) + 1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)}} \]

      6. lower-neg.f32 77.4

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      8. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      9. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      10. lower-fma.f32 77.4

          \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

   10. Applied rewrites 77.4%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \left(\frac{1 - ux}{maxCos} + ux\right) \cdot maxCos, 1\right)}} \]

   11. Taylor expanded in maxCos around 0

       \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(1 - ux\right) \cdot \left(ux - 1\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 75.2%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux - 1, 1 - ux, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.025200000032782555:\\ \;\;\;\;1 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\mathsf{fma}\left(ux - 1, 1 - ux, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<=
        (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0))))
        0.025200000032782555)
     (* 1.0 (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (* 1.0 (sqrt (fma (- ux 1.0) (- 1.0 ux) 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0252

   1. Initial program 37.7%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.3%

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in ux around 0

      \[\leadsto 1 \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.6%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]

   if 0.0252 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 89.4%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around inf

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(maxCos \cdot \left(\left(ux + \frac{1}{maxCos}\right) - \frac{ux}{maxCos}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.3%

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]
   9. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right) + 1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)}} \]

      6. lower-neg.f32 77.4

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      8. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      9. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      10. lower-fma.f32 77.4

          \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

   10. Applied rewrites 77.4%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \left(\frac{1 - ux}{maxCos} + ux\right) \cdot maxCos, 1\right)}} \]

   11. Taylor expanded in maxCos around 0

       \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(1 - ux\right) \cdot \left(ux - 1\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 75.2%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux - 1, 1 - ux, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-maxCos, \mathsf{fma}\left(\mathsf{fma}\left(-2, ux, 2\right), ux, \left(maxCos \cdot ux\right) \cdot ux\right), \left(2 - ux\right) \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) (PI)))
  (sqrt
   (fma
    (- maxCos)
    (fma (fma -2.0 ux 2.0) ux (* (* maxCos ux) ux))
    (* (- 2.0 ux) ux)))))

Derivation

1. Initial program 54.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux}} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{maxCos \cdot \left(-1 \cdot \left(maxCos \cdot {ux}^{2}\right) + -1 \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right) + \color{blue}{ux \cdot \left(2 - ux\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-maxCos, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-2, ux, 2\right), ux, \left(maxCos \cdot ux\right) \cdot ux\right)}, \left(2 - ux\right) \cdot ux\right)} \]
9. Add Preprocessing

Alternative 8: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(2 - ux, ux, \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos\right) \cdot -2\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) (PI)))
  (sqrt (fma (- 2.0 ux) ux (* (* (* (- 1.0 ux) ux) maxCos) -2.0)))))

Derivation

1. Initial program 54.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux}} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right) + \color{blue}{ux \cdot \left(2 - ux\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 98.3%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(2 - ux, \color{blue}{ux}, \left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos\right) \cdot -2\right)} \]
9. Add Preprocessing

Alternative 9: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), 2 \cdot uy, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sin (fma (- (PI)) (* 2.0 uy) (/ (PI) 2.0))) (sqrt (* (- 2.0 ux) ux))))

Derivation

1. Initial program 54.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux}} \]
6. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   2. cos-neg-rev N/A

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

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

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   4. lower-sin.f32 N/A

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   5. lift-*.f32 N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   6. *-commutative N/A

      \[\leadsto \sin \left(\left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \left(uy \cdot 2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   8. lower-fma.f32 N/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right), uy \cdot 2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   9. lower-neg.f32 N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(\color{blue}{-\mathsf{PI}\left(\right)}, uy \cdot 2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   10. lift-*.f32 N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), \color{blue}{uy \cdot 2}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   11. *-commutative N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), \color{blue}{2 \cdot uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   12. lower-*.f32 N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), \color{blue}{2 \cdot uy}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   13. lift-PI.f32 N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), 2 \cdot uy, \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

   14. lower-/.f32 99.1

       \[\leadsto \sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), 2 \cdot uy, \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

7. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), 2 \cdot uy, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \]

8. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), 2 \cdot uy, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 93.5%

    \[\leadsto \sin \left(\mathsf{fma}\left(-\mathsf{PI}\left(\right), 2 \cdot uy, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot \color{blue}{ux}} \]
11. Add Preprocessing

Alternative 10: 92.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (cos (* (* uy 2.0) (PI))) (sqrt (* (- 2.0 ux) ux))))

Derivation

1. Initial program 54.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

5. Applied rewrites 99.1%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux}} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 93.4%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot \color{blue}{ux}} \]
9. Add Preprocessing

Alternative 11: 75.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996600151062012:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \mathsf{fma}\left(maxCos, ux, 1 - ux\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9996600151062012)
     (sqrt (fma (- ux (fma maxCos ux 1.0)) (fma maxCos ux (- 1.0 ux)) 1.0))
     (* 1.0 (sqrt (* (- (fma -1.0 (- maxCos 1.0) 1.0) maxCos) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9996600151062012f) {
		tmp = sqrtf(fmaf((ux - fmaf(maxCos, ux, 1.0f)), fmaf(maxCos, ux, (1.0f - ux)), 1.0f));
	} else {
		tmp = 1.0f * sqrtf(((fmaf(-1.0f, (maxCos - 1.0f), 1.0f) - maxCos) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9996600151062012))
		tmp = sqrt(fma(Float32(ux - fma(maxCos, ux, Float32(1.0))), fma(maxCos, ux, Float32(Float32(1.0) - ux)), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(fma(Float32(-1.0), Float32(maxCos - Float32(1.0)), Float32(1.0)) - maxCos) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999660015

   1. Initial program 87.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around inf

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(maxCos \cdot \left(\left(ux + \frac{1}{maxCos}\right) - \frac{ux}{maxCos}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.0%

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]
   9. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right) + 1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)}} \]

      6. lower-neg.f32 71.0

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      8. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      9. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      10. lower-fma.f32 71.0

          \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

   10. Applied rewrites 71.0%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \left(\frac{1 - ux}{maxCos} + ux\right) \cdot maxCos, 1\right)}} \]

   11. Taylor expanded in uy around 0

       \[\leadsto \color{blue}{\sqrt{1 + \left(ux - \left(1 + maxCos \cdot ux\right)\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 71.8%

       \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \mathsf{fma}\left(maxCos, ux, 1 - ux\right), 1\right)}} \]

   if 0.999660015 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 33.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 29.4%

      \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around inf

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(maxCos \cdot \left(\left(ux + \frac{1}{maxCos}\right) - \frac{ux}{maxCos}\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.1%

      \[\leadsto 1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]
   9. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      2. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos\right) + 1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)}} \]

      6. lower-neg.f32 30.1

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(\color{blue}{-\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      7. lift-+.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      8. +-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      9. lift-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right), \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

      10. lower-fma.f32 30.1

          \[\leadsto 1 \cdot \sqrt{\mathsf{fma}\left(-\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}, \left(ux + \frac{1 - ux}{maxCos}\right) \cdot maxCos, 1\right)} \]

   10. Applied rewrites 30.1%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \left(\frac{1 - ux}{maxCos} + ux\right) \cdot maxCos, 1\right)}} \]

   11. Taylor expanded in ux around 0

       \[\leadsto 1 \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 72.8%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 65.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* 1.0 (sqrt (* (fma -2.0 maxCos 2.0) ux))))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)))
end

Derivation

1. Initial program 54.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

5. Applied rewrites 45.3%

   \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

6. Taylor expanded in ux around 0

   \[\leadsto 1 \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
7. Step-by-step derivation

8. Applied rewrites 63.4%

   \[\leadsto 1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]
9. Add Preprocessing

Alternative 13: 6.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sqrt{1 - 1} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* 1.0 (sqrt (- 1.0 1.0))))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f * sqrtf((1.0f - 1.0f));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 * sqrt((1.0e0 - 1.0e0))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(1.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) * sqrt((single(1.0) - single(1.0)));
end

Derivation

1. Initial program 54.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

5. Applied rewrites 45.3%

   \[\leadsto \color{blue}{1} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

6. Taylor expanded in ux around 0

   \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{1}} \]
7. Step-by-step derivation

8. Applied rewrites 6.6%

   \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{1}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))