UniformSampleCone, x

Percentage Accurate: 57.6% → 99.0%

Time: 13.9s

Alternatives: 13

Speedup: 3.1×

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

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

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))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

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))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.4%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. +-commutative 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+r- 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. fma-def 55.4%

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

   5. +-commutative 55.4%

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

   6. associate-+r- 55.4%

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

   7. fma-def 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   2. cancel-sign-sub-inv 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   4. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   5. unsub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   7. *-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   8. fma-def 99.2%

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

   9. unpow2 99.2%

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

   10. associate-*l* 99.1%

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

   11. sub-neg 99.1%

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

   12. metadata-eval 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in maxCos around 0 99.1%

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

   1. unpow2 99.1%

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

   2. distribute-rgt-out 99.1%

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

9. Simplified 99.1%

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

    1. *-un-lft-identity 99.1%

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

    2. distribute-lft-out-- 99.2%

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

    3. distribute-lft-in 99.2%

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

    4. *-rgt-identity 99.2%

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

    5. +-commutative 99.2%

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

11. Applied egg-rr 99.2%

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

    1. *-lft-identity 99.2%

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

    2. +-commutative 99.2%

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

13. Simplified 99.2%

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

14. Final simplification 99.2%

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

Alternative 2: 97.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.4%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. +-commutative 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+r- 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. fma-def 55.4%

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

   5. +-commutative 55.4%

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

   6. associate-+r- 55.4%

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

   7. fma-def 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   2. cancel-sign-sub-inv 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   4. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   5. unsub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   7. *-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   8. fma-def 99.2%

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

   9. unpow2 99.2%

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

   10. associate-*l* 99.1%

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

   11. sub-neg 99.1%

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

   12. metadata-eval 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in maxCos around 0 97.7%

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

8. Final simplification 97.7%

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

Alternative 3: 97.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.4%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. +-commutative 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+r- 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. fma-def 55.4%

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

   5. +-commutative 55.4%

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

   6. associate-+r- 55.4%

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

   7. fma-def 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   2. cancel-sign-sub-inv 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   4. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   5. unsub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   7. *-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   8. fma-def 99.2%

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

   9. unpow2 99.2%

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

   10. associate-*l* 99.1%

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

   11. sub-neg 99.1%

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

   12. metadata-eval 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in maxCos around 0 97.7%

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

   1. *-rgt-identity 97.7%

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

   2. distribute-lft-out-- 97.6%

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

9. Applied egg-rr 97.6%

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

10. Final simplification 97.6%

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

Alternative 4: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.500000053056283e-6)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux))))
   (sqrt (* ux (- (fma -2.0 maxCos 2.0) (* ux (pow (+ maxCos -1.0) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.500000053056283e-6f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) - (ux * powf((maxCos + -1.0f), 2.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.500000053056283e-6))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) - Float32(ux * (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.50000005e-6

   1. Initial program 55.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.6%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 55.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 55.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 55.6%

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

      5. +-commutative 55.6%

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

      6. associate-+r- 55.6%

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

      7. fma-def 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in ux around -inf 99.1%

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

      1. +-commutative 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]

      2. metadata-eval 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{\left(-2\right)} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)} \]

      3. cancel-sign-sub-inv 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 - 2 \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)} \]

      4. fma-def 99.1%

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

      5. cancel-sign-sub-inv 99.1%

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

      6. metadata-eval 99.1%

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

      7. +-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. fma-def 99.1%

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

      10. mul-1-neg 99.1%

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

      11. *-commutative 99.1%

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

      12. distribute-rgt-neg-in 99.1%

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

      13. mul-1-neg 99.1%

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

      14. unsub-neg 99.1%

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

      15. unpow2 99.1%

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

      16. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

      1. add-exp-log 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in maxCos around 0 99.0%

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

       1. +-commutative 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

       2. mul-1-neg 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux + \color{blue}{\left(-{ux}^{2}\right)}} \]

       3. sub-neg 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot ux - {ux}^{2}}} \]

       4. unpow2 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - \color{blue}{ux \cdot ux}} \]

       5. distribute-rgt-out-- 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - ux\right)}} \]

   11. Simplified 99.0%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]

   if 1.50000005e-6 < maxCos

   1. Initial program 53.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 53.6%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 53.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 53.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 53.5%

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

      5. +-commutative 53.5%

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

      6. associate-+r- 53.4%

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

      7. fma-def 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in ux around 0 99.2%

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

      1. +-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

      2. cancel-sign-sub-inv 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

      3. metadata-eval 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

      4. mul-1-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

      5. unsub-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

      6. +-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

      7. *-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

      8. fma-def 99.2%

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

      9. unpow2 99.2%

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

      10. associate-*l* 99.1%

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

      11. sub-neg 99.1%

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

      12. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in uy around 0 91.3%

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

      1. cancel-sign-sub-inv 91.3%

         \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + \left(-{ux}^{2}\right) \cdot {\left(maxCos - 1\right)}^{2}}} \]

      2. unpow2 91.3%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \left(-\color{blue}{ux \cdot ux}\right) \cdot {\left(maxCos - 1\right)}^{2}} \]

      3. distribute-rgt-neg-out 91.3%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(ux \cdot \left(-ux\right)\right)} \cdot {\left(maxCos - 1\right)}^{2}} \]

      4. sub-neg 91.3%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \left(ux \cdot \left(-ux\right)\right) \cdot {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}} \]

      5. metadata-eval 91.3%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \left(ux \cdot \left(-ux\right)\right) \cdot {\left(maxCos + \color{blue}{-1}\right)}^{2}} \]

      6. associate-*l* 91.2%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{ux \cdot \left(\left(-ux\right) \cdot {\left(maxCos + -1\right)}^{2}\right)}} \]

      7. distribute-lft-neg-in 91.2%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + ux \cdot \color{blue}{\left(-ux \cdot {\left(maxCos + -1\right)}^{2}\right)}} \]

      8. distribute-lft-in 91.5%

         \[\leadsto \sqrt{\color{blue}{ux \cdot \left(\left(2 + -2 \cdot maxCos\right) + \left(-ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}} \]

      9. +-commutative 91.5%

         \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(-2 \cdot maxCos + 2\right)} + \left(-ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)} \]

      10. *-commutative 91.5%

          \[\leadsto \sqrt{ux \cdot \left(\left(\color{blue}{maxCos \cdot -2} + 2\right) + \left(-ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)} \]

      11. fma-udef 91.5%

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

      12. sub-neg 91.5%

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

   9. Simplified 91.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)}\\ \end{array} \]

Alternative 5: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.500000053056283e-6)
   (* (cos (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (* ux ux))))
   (sqrt (* ux (- (fma -2.0 maxCos 2.0) (* ux (pow (+ maxCos -1.0) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.500000053056283e-6f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((2.0f * ux) - (ux * ux)));
	} else {
		tmp = sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) - (ux * powf((maxCos + -1.0f), 2.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.500000053056283e-6))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	else
		tmp = sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) - Float32(ux * (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.50000005e-6

   1. Initial program 55.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.6%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 55.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 55.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 55.6%

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

      5. +-commutative 55.6%

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

      6. associate-+r- 55.6%

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

      7. fma-def 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in ux around -inf 99.1%

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

      1. +-commutative 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]

      2. metadata-eval 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{\left(-2\right)} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)} \]

      3. cancel-sign-sub-inv 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 - 2 \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)} \]

      4. fma-def 99.1%

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

      5. cancel-sign-sub-inv 99.1%

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

      6. metadata-eval 99.1%

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

      7. +-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. fma-def 99.1%

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

      10. mul-1-neg 99.1%

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

      11. *-commutative 99.1%

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

      12. distribute-rgt-neg-in 99.1%

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

      13. mul-1-neg 99.1%

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

      14. unsub-neg 99.1%

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

      15. unpow2 99.1%

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

      16. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

      1. add-exp-log 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in maxCos around 0 99.0%

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

       1. +-commutative 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

       2. mul-1-neg 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux + \color{blue}{\left(-{ux}^{2}\right)}} \]

       3. sub-neg 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot ux - {ux}^{2}}} \]

       4. *-commutative 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot 2} - {ux}^{2}} \]

       5. unpow2 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2 - \color{blue}{ux \cdot ux}} \]

   11. Simplified 99.0%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{ux \cdot 2 - ux \cdot ux}} \]

   if 1.50000005e-6 < maxCos

   1. Initial program 53.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 53.6%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 53.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 53.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 53.5%

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

      5. +-commutative 53.5%

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

      6. associate-+r- 53.4%

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

      7. fma-def 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in ux around 0 99.2%

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

      1. +-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

      2. cancel-sign-sub-inv 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

      3. metadata-eval 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

      4. mul-1-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

      5. unsub-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

      6. +-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

      7. *-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

      8. fma-def 99.2%

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

      9. unpow2 99.2%

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

      10. associate-*l* 99.1%

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

      11. sub-neg 99.1%

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

      12. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in uy around 0 91.3%

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

      1. cancel-sign-sub-inv 91.3%

         \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + \left(-{ux}^{2}\right) \cdot {\left(maxCos - 1\right)}^{2}}} \]

      2. unpow2 91.3%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \left(-\color{blue}{ux \cdot ux}\right) \cdot {\left(maxCos - 1\right)}^{2}} \]

      3. distribute-rgt-neg-out 91.3%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(ux \cdot \left(-ux\right)\right)} \cdot {\left(maxCos - 1\right)}^{2}} \]

      4. sub-neg 91.3%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \left(ux \cdot \left(-ux\right)\right) \cdot {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}} \]

      5. metadata-eval 91.3%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \left(ux \cdot \left(-ux\right)\right) \cdot {\left(maxCos + \color{blue}{-1}\right)}^{2}} \]

      6. associate-*l* 91.2%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{ux \cdot \left(\left(-ux\right) \cdot {\left(maxCos + -1\right)}^{2}\right)}} \]

      7. distribute-lft-neg-in 91.2%

         \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + ux \cdot \color{blue}{\left(-ux \cdot {\left(maxCos + -1\right)}^{2}\right)}} \]

      8. distribute-lft-in 91.5%

         \[\leadsto \sqrt{\color{blue}{ux \cdot \left(\left(2 + -2 \cdot maxCos\right) + \left(-ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}} \]

      9. +-commutative 91.5%

         \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(-2 \cdot maxCos + 2\right)} + \left(-ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)} \]

      10. *-commutative 91.5%

          \[\leadsto \sqrt{ux \cdot \left(\left(\color{blue}{maxCos \cdot -2} + 2\right) + \left(-ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)} \]

      11. fma-udef 91.5%

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

      12. sub-neg 91.5%

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

   9. Simplified 91.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)}\\ \end{array} \]

Alternative 6: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) + \left(ux \cdot ux\right) \cdot \left(-1 - maxCos \cdot \left(maxCos + -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.500000053056283e-6)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux))))
   (sqrt
    (+
     (* ux (fma maxCos -2.0 2.0))
     (* (* ux ux) (- -1.0 (* maxCos (+ maxCos -2.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.500000053056283e-6f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf(((ux * fmaf(maxCos, -2.0f, 2.0f)) + ((ux * ux) * (-1.0f - (maxCos * (maxCos + -2.0f))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.500000053056283e-6))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))) + Float32(Float32(ux * ux) * Float32(Float32(-1.0) - Float32(maxCos * Float32(maxCos + Float32(-2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.50000005e-6

   1. Initial program 55.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.6%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 55.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 55.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 55.6%

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

      5. +-commutative 55.6%

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

      6. associate-+r- 55.6%

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

      7. fma-def 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in ux around -inf 99.1%

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

      1. +-commutative 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]

      2. metadata-eval 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{\left(-2\right)} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)} \]

      3. cancel-sign-sub-inv 99.1%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 - 2 \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)} \]

      4. fma-def 99.1%

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

      5. cancel-sign-sub-inv 99.1%

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

      6. metadata-eval 99.1%

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

      7. +-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. fma-def 99.1%

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

      10. mul-1-neg 99.1%

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

      11. *-commutative 99.1%

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

      12. distribute-rgt-neg-in 99.1%

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

      13. mul-1-neg 99.1%

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

      14. unsub-neg 99.1%

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

      15. unpow2 99.1%

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

      16. distribute-rgt-neg-in 99.1%

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

   6. Simplified 99.1%

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

      1. add-exp-log 96.3%

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

   8. Applied egg-rr 96.3%

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

   9. Taylor expanded in maxCos around 0 99.0%

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

       1. +-commutative 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

       2. mul-1-neg 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux + \color{blue}{\left(-{ux}^{2}\right)}} \]

       3. sub-neg 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot ux - {ux}^{2}}} \]

       4. unpow2 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - \color{blue}{ux \cdot ux}} \]

       5. distribute-rgt-out-- 99.0%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - ux\right)}} \]

   11. Simplified 99.0%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]

   if 1.50000005e-6 < maxCos

   1. Initial program 53.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 53.6%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 53.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 53.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 53.5%

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

      5. +-commutative 53.5%

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

      6. associate-+r- 53.4%

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

      7. fma-def 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in ux around 0 99.2%

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

      1. +-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

      2. cancel-sign-sub-inv 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

      3. metadata-eval 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

      4. mul-1-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

      5. unsub-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

      6. +-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

      7. *-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

      8. fma-def 99.2%

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

      9. unpow2 99.2%

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

      10. associate-*l* 99.1%

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

      11. sub-neg 99.1%

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

      12. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in maxCos around 0 99.1%

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

      1. unpow2 99.1%

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

      2. distribute-rgt-out 99.1%

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

   9. Simplified 99.1%

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

   10. Taylor expanded in uy around 0 91.3%

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

       1. fma-neg 91.5%

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

       2. +-commutative 91.5%

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

       3. *-commutative 91.5%

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

       4. fma-udef 91.5%

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

       5. fma-neg 91.3%

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

       6. unpow2 91.3%

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

       7. sub-neg 91.3%

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

       8. metadata-eval 91.3%

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

       9. +-commutative 91.3%

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

   12. Simplified 91.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) + \left(ux \cdot ux\right) \cdot \left(-1 - maxCos \cdot \left(maxCos + -2\right)\right)}\\ \end{array} \]

Alternative 7: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0008999999845400453:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) + \left(ux \cdot ux\right) \cdot \left(-1 - maxCos \cdot \left(maxCos + -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0008999999845400453)
   (sqrt
    (+
     (* ux (fma maxCos -2.0 2.0))
     (* (* ux ux) (- -1.0 (* maxCos (+ maxCos -2.0))))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0008999999845400453f) {
		tmp = sqrtf(((ux * fmaf(maxCos, -2.0f, 2.0f)) + ((ux * ux) * (-1.0f - (maxCos * (maxCos + -2.0f))))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0008999999845400453))
		tmp = sqrt(Float32(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))) + Float32(Float32(ux * ux) * Float32(Float32(-1.0) - Float32(maxCos * Float32(maxCos + Float32(-2.0)))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 8.99999985e-4

   1. Initial program 56.7%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.7%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 56.7%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 56.7%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 56.7%

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

      5. +-commutative 56.7%

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

      6. associate-+r- 56.6%

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

      7. fma-def 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in ux around 0 99.5%

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

      1. +-commutative 99.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

      2. cancel-sign-sub-inv 99.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

      3. metadata-eval 99.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

      4. mul-1-neg 99.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

      5. unsub-neg 99.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

      6. +-commutative 99.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

      7. *-commutative 99.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

      8. fma-def 99.5%

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

      9. unpow2 99.5%

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

      10. associate-*l* 99.5%

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

      11. sub-neg 99.5%

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

      12. metadata-eval 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in maxCos around 0 99.5%

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

      1. unpow2 99.5%

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

      2. distribute-rgt-out 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in uy around 0 97.4%

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

       1. fma-neg 97.4%

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

       2. +-commutative 97.4%

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

       3. *-commutative 97.4%

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

       4. fma-udef 97.4%

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

       5. fma-neg 97.4%

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

       6. unpow2 97.4%

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

       7. sub-neg 97.4%

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

       8. metadata-eval 97.4%

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

       9. +-commutative 97.4%

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

   12. Simplified 97.4%

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

   if 8.99999985e-4 < uy

   1. Initial program 52.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 52.5%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 52.5%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 52.4%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 52.4%

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

      5. +-commutative 52.4%

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

      6. associate-+r- 52.4%

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

      7. fma-def 52.4%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in ux around 0 43.1%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)}} \]

   5. Taylor expanded in maxCos around 0 77.7%

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

      1. *-commutative 77.7%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot 2}} \]

   7. Simplified 77.7%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0008999999845400453:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) + \left(ux \cdot ux\right) \cdot \left(-1 - maxCos \cdot \left(maxCos + -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 8: 80.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.4%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. +-commutative 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+r- 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. fma-def 55.4%

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

   5. +-commutative 55.4%

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

   6. associate-+r- 55.4%

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

   7. fma-def 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   2. cancel-sign-sub-inv 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   4. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   5. unsub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   7. *-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   8. fma-def 99.2%

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

   9. unpow2 99.2%

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

   10. associate-*l* 99.1%

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

   11. sub-neg 99.1%

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

   12. metadata-eval 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in maxCos around 0 99.1%

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

   1. unpow2 99.1%

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

   2. distribute-rgt-out 99.1%

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

9. Simplified 99.1%

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

10. Taylor expanded in uy around 0 81.6%

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

    1. fma-neg 81.6%

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

    2. +-commutative 81.6%

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

    3. *-commutative 81.6%

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

    4. fma-udef 81.6%

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

    5. fma-neg 81.6%

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

    6. unpow2 81.6%

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

    7. sub-neg 81.6%

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

    8. metadata-eval 81.6%

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

    9. +-commutative 81.6%

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

12. Simplified 81.6%

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

13. Final simplification 81.6%

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

Alternative 9: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right) - {ux}^{2}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (- (* ux (+ 2.0 (* maxCos -2.0))) (pow ux 2.0))))

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(((ux * (2.0e0 + (maxcos * (-2.0e0)))) - (ux ** 2.0e0)))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))) - (ux ^ Float32(2.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt(((ux * (single(2.0) + (maxCos * single(-2.0)))) - (ux ^ single(2.0))));
end

Derivation

1. Initial program 55.4%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. +-commutative 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+r- 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. fma-def 55.4%

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

   5. +-commutative 55.4%

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

   6. associate-+r- 55.4%

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

   7. fma-def 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   2. cancel-sign-sub-inv 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   4. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   5. unsub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   7. *-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   8. fma-def 99.2%

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

   9. unpow2 99.2%

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

   10. associate-*l* 99.1%

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

   11. sub-neg 99.1%

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

   12. metadata-eval 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in maxCos around 0 97.7%

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

8. Taylor expanded in uy around 0 80.3%

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2}}} \]

9. Final simplification 80.3%

   \[\leadsto \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right) - {ux}^{2}} \]

Alternative 10: 74.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00039000000106170774:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00039000000106170774)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt (- 1.0 (pow (- 1.0 ux) 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00039000000106170774f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f - powf((1.0f - ux), 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00039000000106170774e0) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 - ux) ** 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00039000000106170774))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - (Float32(Float32(1.0) - ux) ^ Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00039000000106170774))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) - ux) ^ single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.90000001e-4

   1. Initial program 38.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 38.6%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 38.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 38.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 38.6%

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

      5. +-commutative 38.6%

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

      6. associate-+r- 38.6%

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

      7. fma-def 38.6%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in uy around 0 35.2%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   5. Taylor expanded in ux around 0 76.0%

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

   if 3.90000001e-4 < ux

   1. Initial program 91.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 91.6%

         \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. +-commutative 91.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+r- 91.6%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. fma-def 91.6%

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

      5. +-commutative 91.6%

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

      6. associate-+r- 91.6%

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

      7. fma-def 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in uy around 0 78.5%

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   5. Taylor expanded in maxCos around 0 74.6%

      \[\leadsto \sqrt{\color{blue}{1 - {\left(1 - ux\right)}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00039000000106170774:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \end{array} \]

Alternative 11: 79.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.4%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. +-commutative 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+r- 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. fma-def 55.4%

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

   5. +-commutative 55.4%

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

   6. associate-+r- 55.4%

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

   7. fma-def 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   2. cancel-sign-sub-inv 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(-2\right) \cdot maxCos\right)} + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   3. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{-2} \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)} \]

   4. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right)}} \]

   5. unsub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}}} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   7. *-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{maxCos \cdot -2} + 2\right) - {ux}^{2} \cdot {\left(maxCos - 1\right)}^{2}} \]

   8. fma-def 99.2%

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

   9. unpow2 99.2%

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

   10. associate-*l* 99.1%

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

   11. sub-neg 99.1%

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

   12. metadata-eval 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in maxCos around 0 97.7%

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

8. Taylor expanded in uy around 0 80.3%

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

   1. fma-neg 80.3%

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

   2. +-commutative 80.3%

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

   3. *-commutative 80.3%

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

   4. fma-udef 80.3%

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

   5. fma-neg 80.3%

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

   6. unpow2 80.3%

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

   7. distribute-lft-out-- 80.3%

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

   8. fma-udef 80.3%

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(maxCos \cdot -2 + 2\right)} - ux\right)} \]

   9. *-commutative 80.3%

      \[\leadsto \sqrt{ux \cdot \left(\left(\color{blue}{-2 \cdot maxCos} + 2\right) - ux\right)} \]

   10. fma-def 80.3%

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

10. Simplified 80.3%

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

11. Final simplification 80.3%

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

Alternative 12: 64.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
end function

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
end

Derivation

1. Initial program 55.4%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. +-commutative 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+r- 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. fma-def 55.4%

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

   5. +-commutative 55.4%

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

   6. associate-+r- 55.4%

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

   7. fma-def 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in uy around 0 48.9%

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

5. Taylor expanded in ux around 0 66.7%

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

6. Final simplification 66.7%

   \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

Alternative 13: 62.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((2.0e0 * ux))
end function

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(2.0) * ux));
end

Derivation

1. Initial program 55.4%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.4%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. +-commutative 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+r- 55.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. fma-def 55.4%

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

   5. +-commutative 55.4%

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

   6. associate-+r- 55.4%

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

   7. fma-def 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in uy around 0 48.9%

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

5. Taylor expanded in ux around 0 66.7%

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

6. Taylor expanded in maxCos around 0 63.8%

   \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
7. Step-by-step derivation

   1. *-commutative 63.8%

      \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

8. Simplified 63.8%

   \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

9. Final simplification 63.8%

   \[\leadsto \sqrt{2 \cdot ux} \]

Reproduce

herbie shell --seed 2023293 
(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)))))))