UniformSampleCone, x

Percentage Accurate: 57.2% → 99.5%

Time: 25.9s

Alternatives: 15

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: 57.2% 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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \langle \left( \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, -1 + \left(ux - ux \cdot maxCos\right), 1\right)} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (*
    (cos (* uy (* 2.0 PI)))
    (sqrt
     (fma (- (fma ux maxCos 1.0) ux) (+ -1.0 (- ux (* ux maxCos))) 1.0))))))

C

float code(float ux, float uy, float maxCos) {
	double tmp = cos((((double) uy) * (2.0 * ((double) M_PI)))) * sqrt(fma((fma(ux, maxCos, 1.0) - ((double) ux)), (-1.0 + (((double) ux) - (((double) ux) * ((double) maxCos)))), 1.0));
	return (float) tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float64(cos(Float64(Float64(uy) * Float64(2.0 * pi))) * sqrt(fma(Float64(fma(ux, maxCos, 1.0) - Float64(ux)), Float64(-1.0 + Float64(Float64(ux) - Float64(Float64(ux) * Float64(maxCos)))), 1.0)))
	return Float32(tmp)
end

Derivation

1. Initial program 99.5%

   \[\langle \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, -1 + \left(ux - ux \cdot maxCos\right), 1\right)} \rangle_{\text{binary64}} \]

2. Final simplification 99.5%

   \[\leadsto \langle \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, -1 + \left(ux - ux \cdot maxCos\right), 1\right)} \rangle_{\text{binary64}} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (*
    (cos (* 2.0 (* uy PI)))
    (sqrt (- 1.0 (pow (- (fma ux maxCos 1.0) ux) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	double tmp = cos((2.0 * (((double) uy) * ((double) M_PI)))) * sqrt((1.0 - pow((fma(ux, maxCos, 1.0) - ((double) ux)), 2.0)));
	return (float) tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float64(cos(Float64(2.0 * Float64(Float64(uy) * pi))) * sqrt(Float64(1.0 - (Float64(fma(ux, maxCos, 1.0) - Float64(ux)) ^ 2.0))))
	return Float32(tmp)
end

Derivation

1. Initial program 99.5%

   \[\langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos, 1 - ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)} \rangle_{\text{binary64}} \]
2. Step-by-step derivation

   1. sub-neg 99.5%

      \[\leadsto \langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 + \left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)} \rangle_{\text{binary64}} \]

   2. pow2 99.5%

      \[\leadsto \langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 + \left(-{\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{2}\right)} \rangle_{\text{binary64}} \]

   3. fma-udef 99.5%

      \[\leadsto \langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 + \left(-{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}^{2}\right)} \rangle_{\text{binary64}} \]

   4. associate-+r- 99.5%

      \[\leadsto \langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 + \left(-{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}^{2}\right)} \rangle_{\text{binary64}} \]

   5. fma-udef 99.5%

      \[\leadsto \langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 + \left(-{\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}\right)} \rangle_{\text{binary64}} \]

3. Applied egg-rr 99.5%

   \[\leadsto \langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 + \left(-{\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}\right)} \rangle_{\text{binary64}} \]
4. Step-by-step derivation

   1. unsub-neg 99.5%

      \[\leadsto \langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}} \rangle_{\text{binary64}} \]

5. Simplified 99.5%

   \[\leadsto \langle \cos \left(\left(\pi \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}} \rangle_{\text{binary64}} \]

6. Final simplification 99.5%

   \[\leadsto \langle \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}} \rangle_{\text{binary64}} \]

Alternative 3: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \langle \left( \sqrt{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* PI (* uy 2.0)))
  (cast
   (!
    :precision
    binary64
    (sqrt (- 1.0 (pow (- (fma ux maxCos 1.0) ux) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	double tmp = sqrt((1.0 - pow((fma(ux, maxCos, 1.0) - ((double) ux)), 2.0)));
	return cosf((((float) M_PI) * (uy * 2.0f))) * ((float) tmp);
}

Julia

function code(ux, uy, maxCos)
	tmp = sqrt(Float64(1.0 - (Float64(fma(ux, maxCos, 1.0) - Float64(ux)) ^ 2.0)))
	return Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * Float32(tmp))
end

Derivation

1. Initial program 58.6%

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

   1. rewrite-binary32/binary64 99.3%

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \langle \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \rangle_{\text{binary64}}} \]

3. Applied rewrite-once 99.3%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\langle \color{blue}{\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \rangle_{\text{binary64}}} \]
4. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \langle \sqrt{1 - \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \rangle_{\text{binary64}} \]

   2. associate-+r- 99.3%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \langle \sqrt{1 - \left(\left(ux \cdot maxCos + 1\right) - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \rangle_{\text{binary64}} \]

   3. fma-udef 99.3%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \langle \sqrt{1 - \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \rangle_{\text{binary64}} \]

   4. +-commutative 99.3%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \langle \sqrt{1 - \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)} \rangle_{\text{binary64}} \]

   5. associate-+r- 99.3%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \langle \sqrt{1 - \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot \left(\left(ux \cdot maxCos + 1\right) - ux\right)} \rangle_{\text{binary64}} \]

   6. fma-udef 99.3%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \langle \sqrt{1 - \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)} \rangle_{\text{binary64}} \]

5. Simplified 99.3%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\langle \color{blue}{\sqrt{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}}} \rangle_{\text{binary64}}} \]

6. Final simplification 99.3%

   \[\leadsto \cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \langle \sqrt{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)}^{2}} \rangle_{\text{binary64}} \]

Alternative 4: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. associate--l+ 99.2%

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

   3. mul-1-neg 99.2%

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

   4. sub-neg 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-neg-in 99.2%

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

   7. mul-1-neg 99.2%

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

   8. metadata-eval 99.2%

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

   9. +-commutative 99.2%

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

   10. mul-1-neg 99.2%

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

   11. sub-neg 99.2%

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

   12. unpow2 99.2%

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

   13. *-commutative 99.2%

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

   14. sub-neg 99.2%

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

   15. metadata-eval 99.2%

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

6. Simplified 99.2%

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

   1. add-exp-log_binary32 99.2%

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

8. Applied rewrite-once 99.2%

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

9. Final simplification 99.2%

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

Alternative 5: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. associate--l+ 99.2%

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

   3. mul-1-neg 99.2%

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

   4. sub-neg 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-neg-in 99.2%

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

   7. mul-1-neg 99.2%

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

   8. metadata-eval 99.2%

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

   9. +-commutative 99.2%

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

   10. mul-1-neg 99.2%

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

   11. sub-neg 99.2%

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

   12. unpow2 99.2%

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

   13. *-commutative 99.2%

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

   14. sub-neg 99.2%

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

   15. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in uy around inf 99.2%

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

   1. fma-def 99.2%

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

   2. cancel-sign-sub-inv 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

   5. unpow2 99.2%

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

   6. sub-neg 99.2%

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

   7. metadata-eval 99.2%

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

   8. +-commutative 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 6: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0013000000035390258:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, \left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0013000000035390258)
   (sqrt
    (fma
     ux
     (+ 2.0 (* maxCos -2.0))
     (* (* ux ux) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- (- 2.0 maxCos) maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0013000000035390258f) {
		tmp = sqrtf(fmaf(ux, (2.0f + (maxCos * -2.0f)), ((ux * ux) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((2.0f - maxCos) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0013000000035390258))
		tmp = sqrt(fma(ux, Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - maxCos) - maxCos))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.0013

   1. Initial program 59.1%

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

      1. associate-*l* 59.1%

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

      2. sub-neg 59.1%

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

      3. +-commutative 59.1%

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

      4. distribute-rgt-neg-in 59.1%

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

      5. fma-def 59.0%

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

      6. +-commutative 59.0%

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

      7. associate-+r- 59.1%

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

      8. fma-def 59.1%

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

      9. distribute-neg-in 59.1%

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

      10. unsub-neg 59.1%

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

      11. neg-sub0 59.1%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

      12. associate--r- 59.1%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

      13. metadata-eval 59.1%

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

      14. associate-+r- 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in ux around 0 99.6%

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

      1. fma-def 99.6%

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

      2. associate--l+ 99.6%

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

      3. mul-1-neg 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. +-commutative 99.6%

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

      10. mul-1-neg 99.6%

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

      11. sub-neg 99.6%

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

      12. unpow2 99.6%

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

      13. *-commutative 99.6%

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

      14. sub-neg 99.6%

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

      15. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 97.8%

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

      1. fma-def 97.8%

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

      2. cancel-sign-sub-inv 97.8%

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

      3. metadata-eval 97.8%

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

      4. *-commutative 97.8%

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

      5. unpow2 97.8%

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

      6. sub-neg 97.8%

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

      7. metadata-eval 97.8%

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

      8. +-commutative 97.8%

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

   9. Simplified 97.8%

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

   if 0.0013 < (*.f32 uy 2)

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. sub-neg 57.3%

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

      3. +-commutative 57.3%

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

      4. distribute-rgt-neg-in 57.3%

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

      5. fma-def 57.2%

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

      6. +-commutative 57.2%

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

      7. associate-+r- 57.2%

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

      8. fma-def 57.2%

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

      9. distribute-neg-in 57.2%

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

      10. unsub-neg 57.2%

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

      11. neg-sub0 57.2%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

      12. associate--r- 57.2%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

      13. metadata-eval 57.2%

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

      14. associate-+r- 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in ux around 0 78.6%

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

      1. sub-neg 78.6%

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

      2. mul-1-neg 78.6%

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

      3. sub-neg 78.6%

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

      4. metadata-eval 78.6%

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

      5. +-commutative 78.6%

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

      6. distribute-neg-in 78.6%

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

      7. metadata-eval 78.6%

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

      8. sub-neg 78.6%

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

      9. associate-+r- 78.7%

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

      10. metadata-eval 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. unsub-neg 78.7%

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

   8. Simplified 78.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0013000000035390258:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, \left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \end{array} \]

Alternative 7: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 9.500000305706635 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, \left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\left(ux + ux\right) - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 9.500000305706635e-5)
   (sqrt
    (fma
     ux
     (+ 2.0 (* maxCos -2.0))
     (* (* ux ux) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (* (cos (* PI (* uy 2.0))) (sqrt (- (+ ux ux) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 9.500000305706635e-5f) {
		tmp = sqrtf(fmaf(ux, (2.0f + (maxCos * -2.0f)), ((ux * ux) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf(((ux + ux) - (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(9.500000305706635e-5))
		tmp = sqrt(fma(ux, Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(Float32(ux + ux) - Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 9.50000031e-5

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. sub-neg 59.2%

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

      3. +-commutative 59.2%

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

      4. distribute-rgt-neg-in 59.2%

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

      5. fma-def 59.1%

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

      6. +-commutative 59.1%

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

      7. associate-+r- 59.2%

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

      8. fma-def 59.2%

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

      9. distribute-neg-in 59.2%

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

      10. unsub-neg 59.2%

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

      11. neg-sub0 59.2%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

      12. associate--r- 59.2%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

      13. metadata-eval 59.2%

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

      14. associate-+r- 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in ux around 0 99.7%

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

      1. fma-def 99.7%

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

      2. associate--l+ 99.7%

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

      3. mul-1-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. mul-1-neg 99.7%

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

      8. metadata-eval 99.7%

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

      9. +-commutative 99.7%

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

      10. mul-1-neg 99.7%

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

      11. sub-neg 99.7%

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

      12. unpow2 99.7%

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

      13. *-commutative 99.7%

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

      14. sub-neg 99.7%

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

      15. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in uy around 0 99.6%

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

      1. fma-def 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. unpow2 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

      8. +-commutative 99.7%

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

   9. Simplified 99.7%

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

   if 9.50000031e-5 < (*.f32 uy 2)

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

      2. sub-neg 57.7%

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

      3. +-commutative 57.7%

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

      4. distribute-rgt-neg-in 57.7%

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

      5. fma-def 57.7%

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

      6. +-commutative 57.7%

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

      7. associate-+r- 57.6%

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

      8. fma-def 57.6%

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

      9. distribute-neg-in 57.6%

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

      10. unsub-neg 57.6%

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

      11. neg-sub0 57.6%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

      12. associate--r- 57.6%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

      13. metadata-eval 57.6%

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

      14. associate-+r- 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around 0 98.5%

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

      1. fma-def 98.5%

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

      2. associate--l+ 98.6%

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

      3. mul-1-neg 98.6%

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

      4. sub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. distribute-neg-in 98.6%

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

      7. mul-1-neg 98.6%

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

      8. metadata-eval 98.6%

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

      9. +-commutative 98.6%

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

      10. mul-1-neg 98.6%

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

      11. sub-neg 98.6%

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

      12. unpow2 98.6%

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

      13. *-commutative 98.6%

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

      14. sub-neg 98.6%

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

      15. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

      1. add-exp-log_binary32 98.6%

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

   8. Applied rewrite-once 98.6%

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

   9. Taylor expanded in maxCos around 0 94.6%

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

       1. associate-*r* 94.6%

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

       2. +-commutative 94.6%

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

       3. mul-1-neg 94.6%

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

       4. unsub-neg 94.6%

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

       5. count-2 94.6%

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

       6. unpow2 94.6%

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

   11. Simplified 94.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 9.500000305706635 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, \left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\left(ux + ux\right) - ux \cdot ux}\\ \end{array} \]

Alternative 8: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0013000000035390258:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, \left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0013000000035390258)
   (sqrt
    (fma
     ux
     (+ 2.0 (* maxCos -2.0))
     (* (* ux ux) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0013000000035390258f) {
		tmp = sqrtf(fmaf(ux, (2.0f + (maxCos * -2.0f)), ((ux * ux) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
	} 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 (Float32(uy * Float32(2.0)) <= Float32(0.0013000000035390258))
		tmp = sqrt(fma(ux, Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))));
	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 (*.f32 uy 2) < 0.0013

   1. Initial program 59.1%

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

      1. associate-*l* 59.1%

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

      2. sub-neg 59.1%

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

      3. +-commutative 59.1%

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

      4. distribute-rgt-neg-in 59.1%

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

      5. fma-def 59.0%

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

      6. +-commutative 59.0%

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

      7. associate-+r- 59.1%

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

      8. fma-def 59.1%

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

      9. distribute-neg-in 59.1%

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

      10. unsub-neg 59.1%

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

      11. neg-sub0 59.1%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

      12. associate--r- 59.1%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

      13. metadata-eval 59.1%

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

      14. associate-+r- 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in ux around 0 99.6%

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

      1. fma-def 99.6%

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

      2. associate--l+ 99.6%

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

      3. mul-1-neg 99.6%

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

      4. sub-neg 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. +-commutative 99.6%

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

      10. mul-1-neg 99.6%

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

      11. sub-neg 99.6%

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

      12. unpow2 99.6%

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

      13. *-commutative 99.6%

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

      14. sub-neg 99.6%

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

      15. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 97.8%

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

      1. fma-def 97.8%

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

      2. cancel-sign-sub-inv 97.8%

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

      3. metadata-eval 97.8%

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

      4. *-commutative 97.8%

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

      5. unpow2 97.8%

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

      6. sub-neg 97.8%

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

      7. metadata-eval 97.8%

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

      8. +-commutative 97.8%

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

   9. Simplified 97.8%

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

   if 0.0013 < (*.f32 uy 2)

   1. Initial program 57.3%

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

      1. associate-*l* 57.3%

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

      2. sub-neg 57.3%

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

      3. +-commutative 57.3%

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

      4. distribute-rgt-neg-in 57.3%

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

      5. fma-def 57.2%

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

      6. +-commutative 57.2%

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

      7. associate-+r- 57.2%

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

      8. fma-def 57.2%

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

      9. distribute-neg-in 57.2%

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

      10. unsub-neg 57.2%

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

      11. neg-sub0 57.2%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

      12. associate--r- 57.2%

          \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

      13. metadata-eval 57.2%

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

      14. associate-+r- 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in ux around 0 78.6%

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

   5. Taylor expanded in maxCos around 0 75.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0013000000035390258:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, \left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 9: 79.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. associate--l+ 99.2%

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

   3. mul-1-neg 99.2%

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

   4. sub-neg 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-neg-in 99.2%

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

   7. mul-1-neg 99.2%

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

   8. metadata-eval 99.2%

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

   9. +-commutative 99.2%

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

   10. mul-1-neg 99.2%

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

   11. sub-neg 99.2%

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

   12. unpow2 99.2%

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

   13. *-commutative 99.2%

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

   14. sub-neg 99.2%

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

   15. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in uy around 0 81.8%

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

   1. fma-def 81.8%

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

   2. cancel-sign-sub-inv 81.8%

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

   3. metadata-eval 81.8%

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

   4. *-commutative 81.8%

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

   5. unpow2 81.8%

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

   6. sub-neg 81.8%

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

   7. metadata-eval 81.8%

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

   8. +-commutative 81.8%

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

9. Simplified 81.8%

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

10. Final simplification 81.8%

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

Alternative 10: 64.7% 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 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in uy around 0 51.7%

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

   1. associate--r+ 51.8%

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

   2. sub-neg 51.8%

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

   3. metadata-eval 51.8%

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

   4. +-commutative 51.8%

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

   5. associate-+r- 51.9%

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

   6. *-commutative 51.9%

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

   7. flip-+ 52.1%

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

   8. clear-num 51.7%

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

   9. *-commutative 51.7%

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

   10. associate--r- 51.6%

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

   11. *-commutative 51.6%

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

   12. metadata-eval 51.6%

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

   13. pow2 51.6%

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

6. Applied egg-rr 51.6%

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

7. Taylor expanded in ux around 0 66.2%

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

   1. associate--l+ 66.1%

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

   2. mul-1-neg 66.1%

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

   3. sub-neg 66.1%

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

   4. metadata-eval 66.1%

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

   5. +-commutative 66.1%

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

   6. distribute-neg-in 66.1%

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

   7. metadata-eval 66.1%

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

   8. sub-neg 66.1%

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

   9. associate--l- 66.1%

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

9. Simplified 66.1%

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

10. Taylor expanded in ux around 0 66.1%

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

11. Final simplification 66.1%

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

Alternative 11: 75.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((ux + ux) - (ux * 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(((ux + ux) - (ux * ux)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in uy around 0 51.7%

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

   1. associate--r+ 51.8%

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

   2. sub-neg 51.8%

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

   3. metadata-eval 51.8%

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

   4. +-commutative 51.8%

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

   5. associate-+r- 51.9%

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

   6. *-commutative 51.9%

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

   7. flip-+ 52.1%

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

   8. clear-num 51.7%

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

   9. *-commutative 51.7%

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

   10. associate--r- 51.6%

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

   11. *-commutative 51.6%

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

   12. metadata-eval 51.6%

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

   13. pow2 51.6%

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

6. Applied egg-rr 51.6%

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

7. Taylor expanded in maxCos around 0 50.1%

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

8. Taylor expanded in ux around 0 77.1%

   \[\leadsto \sqrt{\color{blue}{-1 \cdot {ux}^{2} + 2 \cdot ux}} \]
9. Step-by-step derivation

   1. +-commutative 77.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

   2. mul-1-neg 77.1%

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

   3. unsub-neg 77.1%

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

   4. count-2 77.1%

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

   5. unpow2 77.1%

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

10. Simplified 77.1%

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

11. Final simplification 77.1%

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

Alternative 12: 25.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot 0.25} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux 0.25)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * 0.25f));
}

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 * 0.25e0))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(0.25)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * single(0.25)));
end

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in uy around 0 51.7%

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

   1. associate--r+ 51.8%

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

   2. sub-neg 51.8%

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

   3. metadata-eval 51.8%

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

   4. +-commutative 51.8%

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

   5. associate-+r- 51.9%

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

   6. *-commutative 51.9%

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

   7. flip-+ 52.1%

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

   8. clear-num 51.7%

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

   9. *-commutative 51.7%

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

   10. associate--r- 51.6%

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

   11. *-commutative 51.6%

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

   12. metadata-eval 51.6%

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

   13. pow2 51.6%

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

6. Applied egg-rr 51.6%

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

7. Taylor expanded in ux around 0 66.2%

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

   1. associate--l+ 66.1%

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

   2. mul-1-neg 66.1%

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

   3. sub-neg 66.1%

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

   4. metadata-eval 66.1%

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

   5. +-commutative 66.1%

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

   6. distribute-neg-in 66.1%

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

   7. metadata-eval 66.1%

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

   8. sub-neg 66.1%

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

   9. associate--l- 66.1%

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

9. Simplified 66.1%

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

11. Applied egg-rr 25.4%

    \[\leadsto \sqrt{ux \cdot \color{blue}{0.25}} \]

12. Final simplification 25.4%

    \[\leadsto \sqrt{ux \cdot 0.25} \]

Alternative 13: 27.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot 0.5} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux 0.5)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * 0.5f));
}

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 * 0.5e0))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(0.5)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * single(0.5)));
end

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in uy around 0 51.7%

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

   1. associate--r+ 51.8%

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

   2. sub-neg 51.8%

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

   3. metadata-eval 51.8%

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

   4. +-commutative 51.8%

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

   5. associate-+r- 51.9%

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

   6. *-commutative 51.9%

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

   7. flip-+ 52.1%

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

   8. clear-num 51.7%

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

   9. *-commutative 51.7%

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

   10. associate--r- 51.6%

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

   11. *-commutative 51.6%

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

   12. metadata-eval 51.6%

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

   13. pow2 51.6%

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

6. Applied egg-rr 51.6%

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

7. Taylor expanded in ux around 0 66.2%

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

   1. associate--l+ 66.1%

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

   2. mul-1-neg 66.1%

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

   3. sub-neg 66.1%

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

   4. metadata-eval 66.1%

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

   5. +-commutative 66.1%

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

   6. distribute-neg-in 66.1%

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

   7. metadata-eval 66.1%

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

   8. sub-neg 66.1%

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

   9. associate--l- 66.1%

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

9. Simplified 66.1%

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

11. Applied egg-rr 27.1%

    \[\leadsto \sqrt{ux \cdot \color{blue}{0.5}} \]

12. Final simplification 27.1%

    \[\leadsto \sqrt{ux \cdot 0.5} \]

Alternative 14: 62.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux + ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (+ ux ux)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux + 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((ux + ux))
end function

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux + ux));
end

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in uy around 0 51.7%

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

   1. associate--r+ 51.8%

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

   2. sub-neg 51.8%

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

   3. metadata-eval 51.8%

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

   4. +-commutative 51.8%

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

   5. associate-+r- 51.9%

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

   6. *-commutative 51.9%

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

   7. flip-+ 52.1%

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

   8. clear-num 51.7%

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

   9. *-commutative 51.7%

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

   10. associate--r- 51.6%

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

   11. *-commutative 51.6%

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

   12. metadata-eval 51.6%

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

   13. pow2 51.6%

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

6. Applied egg-rr 51.6%

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

7. Taylor expanded in ux around 0 66.2%

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

   1. associate--l+ 66.1%

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

   2. mul-1-neg 66.1%

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

   3. sub-neg 66.1%

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

   4. metadata-eval 66.1%

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

   5. +-commutative 66.1%

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

   6. distribute-neg-in 66.1%

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

   7. metadata-eval 66.1%

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

   8. sub-neg 66.1%

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

   9. associate--l- 66.1%

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

9. Simplified 66.1%

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

10. Taylor expanded in maxCos around 0 63.5%

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

    1. count-2 63.5%

       \[\leadsto \sqrt{\color{blue}{ux + ux}} \]

12. Simplified 63.5%

    \[\leadsto \sqrt{\color{blue}{ux + ux}} \]

13. Final simplification 63.5%

    \[\leadsto \sqrt{ux + ux} \]

Alternative 15: 4.5% accurate, 107.3× speedup

Math

\[\begin{array}{l} \\ ux \cdot -8 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* ux -8.0))

C

float code(float ux, float uy, float maxCos) {
	return ux * -8.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * (-8.0e0)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(ux * Float32(-8.0))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ux * single(-8.0);
end

Derivation

1. Initial program 58.6%

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

   1. associate-*l* 58.6%

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

   2. sub-neg 58.6%

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

   3. +-commutative 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

   5. fma-def 58.5%

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

   6. +-commutative 58.5%

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

   7. associate-+r- 58.5%

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

   8. fma-def 58.5%

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

   9. distribute-neg-in 58.5%

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

   10. unsub-neg 58.5%

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

   11. neg-sub0 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(1 - ux\right)\right)} - ux \cdot maxCos, 1\right)} \]

   12. associate--r- 58.5%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(\left(0 - 1\right) + ux\right)} - ux \cdot maxCos, 1\right)} \]

   13. metadata-eval 58.5%

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

   14. associate-+r- 58.6%

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

3. Simplified 58.6%

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

4. Taylor expanded in uy around 0 51.7%

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

   1. associate--r+ 51.8%

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

   2. sub-neg 51.8%

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

   3. metadata-eval 51.8%

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

   4. +-commutative 51.8%

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

   5. associate-+r- 51.9%

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

   6. *-commutative 51.9%

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

   7. flip-+ 52.1%

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

   8. clear-num 51.7%

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

   9. *-commutative 51.7%

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

   10. associate--r- 51.6%

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

   11. *-commutative 51.6%

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

   12. metadata-eval 51.6%

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

   13. pow2 51.6%

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

6. Applied egg-rr 51.6%

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

7. Taylor expanded in ux around 0 66.2%

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

   1. associate--l+ 66.1%

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

   2. mul-1-neg 66.1%

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

   3. sub-neg 66.1%

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

   4. metadata-eval 66.1%

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

   5. +-commutative 66.1%

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

   6. distribute-neg-in 66.1%

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

   7. metadata-eval 66.1%

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

   8. sub-neg 66.1%

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

   9. associate--l- 66.1%

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

9. Simplified 66.1%

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

11. Applied egg-rr 4.4%

    \[\leadsto \color{blue}{-8 \cdot ux} \]
12. Step-by-step derivation

    1. *-commutative 4.4%

       \[\leadsto \color{blue}{ux \cdot -8} \]

13. Simplified 4.4%

    \[\leadsto \color{blue}{ux \cdot -8} \]

14. Final simplification 4.4%

    \[\leadsto ux \cdot -8 \]

Reproduce

herbie shell --seed 2023297 
(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)))))))