UniformSampleCone, x

Percentage Accurate: 57.0% → 99.0%

Time: 14.5s

Alternatives: 12

Speedup: 2.2×

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: 57.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(-1 + e^{\mathsf{log1p}\left({\left(-1 + maxCos\right)}^{2} \cdot \left(-ux\right)\right)}\right) + \left(2 - 2 \cdot maxCos\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (* 2.0 PI)))
  (sqrt
   (*
    ux
    (+
     (+ -1.0 (exp (log1p (* (pow (+ -1.0 maxCos) 2.0) (- ux)))))
     (- 2.0 (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((-1.0f + expf(log1pf((powf((-1.0f + maxCos), 2.0f) * -ux)))) + (2.0f - (2.0f * maxCos)))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(-1.0) + exp(log1p(Float32((Float32(Float32(-1.0) + maxCos) ^ Float32(2.0)) * Float32(-ux))))) + Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))))))
end

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in ux around 0 99.2%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   2. expm1-undefine 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   3. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{-ux \cdot {\left(maxCos - 1\right)}^{2}}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   4. sub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   5. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(maxCos + \color{blue}{-1}\right)}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(-1 + maxCos\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

7. Applied egg-rr 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(-1 + maxCos\right)}^{2}\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

8. Final simplification 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(-1 + e^{\mathsf{log1p}\left({\left(-1 + maxCos\right)}^{2} \cdot \left(-ux\right)\right)}\right) + \left(2 - 2 \cdot maxCos\right)\right)} \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in ux around 0 99.2%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   2. expm1-undefine 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   3. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{-ux \cdot {\left(maxCos - 1\right)}^{2}}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   4. sub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   5. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(maxCos + \color{blue}{-1}\right)}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(-1 + maxCos\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

7. Applied egg-rr 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(-1 + maxCos\right)}^{2}\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

8. Taylor expanded in maxCos around 0 99.2%

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

   1. associate--l+ 99.2%

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

   2. associate-*r* 99.2%

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

   3. neg-mul-1 99.2%

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

   4. *-commutative 99.2%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - \left(ux + maxCos \cdot \left(\left(2 + ux \cdot -2\right) + ux \cdot maxCos\right)\right)\right)} \]
12. Add Preprocessing

Alternative 3: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 - ux \cdot {\left(-1 + maxCos\right)}^{2}\right) - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999873689376e-5)
   (* (sqrt (* ux (- 2.0 ux))) (cos (* 2.0 (* uy PI))))
   (sqrt (* ux (- (- 2.0 (* ux (pow (+ -1.0 maxCos) 2.0))) (* 2.0 maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.999999873689376e-5f) {
		tmp = sqrtf((ux * (2.0f - ux))) * cosf((2.0f * (uy * ((float) M_PI))));
	} else {
		tmp = sqrtf((ux * ((2.0f - (ux * powf((-1.0f + maxCos), 2.0f))) - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999873689376e-5))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(2.0) - Float32(ux * (Float32(Float32(-1.0) + maxCos) ^ Float32(2.0)))) - Float32(Float32(2.0) * maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.999999873689376e-5))
		tmp = sqrt((ux * (single(2.0) - ux))) * cos((single(2.0) * (uy * single(pi))));
	else
		tmp = sqrt((ux * ((single(2.0) - (ux * ((single(-1.0) + maxCos) ^ single(2.0)))) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-5

   1. Initial program 56.3%

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

      1. associate-*l* 56.3%

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

      2. sub-neg 56.3%

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

      3. +-commutative 56.3%

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

      4. distribute-rgt-neg-in 56.3%

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

      5. fma-define 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in ux around 0 99.2%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      2. expm1-undefine 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      3. mul-1-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{-ux \cdot {\left(maxCos - 1\right)}^{2}}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      4. sub-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      5. metadata-eval 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(maxCos + \color{blue}{-1}\right)}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      6. +-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(-1 + maxCos\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   7. Applied egg-rr 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(-1 + maxCos\right)}^{2}\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   8. Taylor expanded in maxCos around 0 98.4%

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

   if 4.99999987e-5 < maxCos

   1. Initial program 61.1%

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

      1. associate-*l* 61.1%

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

      2. sub-neg 61.1%

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

      3. +-commutative 61.1%

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

      4. distribute-rgt-neg-in 61.1%

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

      5. fma-define 61.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in uy around 0 52.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in ux around 0 80.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 - ux \cdot {\left(-1 + maxCos\right)}^{2}\right) - 2 \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in ux around 0 99.2%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   2. expm1-undefine 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   3. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{-ux \cdot {\left(maxCos - 1\right)}^{2}}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   4. sub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   5. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(maxCos + \color{blue}{-1}\right)}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(-1 + maxCos\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

7. Applied egg-rr 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(-1 + maxCos\right)}^{2}\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

8. Taylor expanded in maxCos around 0 98.7%

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

   1. associate--l+ 98.7%

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

   2. mul-1-neg 98.7%

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

   3. *-commutative 98.7%

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

10. Simplified 98.7%

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

11. Final simplification 98.7%

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - \left(ux + maxCos \cdot \left(2 + ux \cdot -2\right)\right)\right)} \]
12. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in ux around 0 99.2%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   2. expm1-undefine 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   3. mul-1-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{-ux \cdot {\left(maxCos - 1\right)}^{2}}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   4. sub-neg 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   5. metadata-eval 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(maxCos + \color{blue}{-1}\right)}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   6. +-commutative 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(-1 + maxCos\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

7. Applied egg-rr 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(-1 + maxCos\right)}^{2}\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

8. Taylor expanded in maxCos around 0 98.7%

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

   1. mul-1-neg 98.7%

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

   2. unsub-neg 98.7%

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

   3. *-commutative 98.7%

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

10. Simplified 98.7%

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

Alternative 6: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - \left(2 \cdot maxCos + ux \cdot \left(1 + maxCos \cdot \left(maxCos - 2\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999873689376e-5)
   (* (sqrt (* ux (- 2.0 ux))) (cos (* 2.0 (* uy PI))))
   (sqrt
    (*
     ux
     (- 2.0 (+ (* 2.0 maxCos) (* ux (+ 1.0 (* maxCos (- maxCos 2.0))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.999999873689376e-5f) {
		tmp = sqrtf((ux * (2.0f - ux))) * cosf((2.0f * (uy * ((float) M_PI))));
	} else {
		tmp = sqrtf((ux * (2.0f - ((2.0f * maxCos) + (ux * (1.0f + (maxCos * (maxCos - 2.0f))))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999873689376e-5))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(Float32(2.0) * maxCos) + Float32(ux * Float32(Float32(1.0) + Float32(maxCos * Float32(maxCos - Float32(2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.999999873689376e-5))
		tmp = sqrt((ux * (single(2.0) - ux))) * cos((single(2.0) * (uy * single(pi))));
	else
		tmp = sqrt((ux * (single(2.0) - ((single(2.0) * maxCos) + (ux * (single(1.0) + (maxCos * (maxCos - single(2.0)))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-5

   1. Initial program 56.3%

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

      1. associate-*l* 56.3%

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

      2. sub-neg 56.3%

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

      3. +-commutative 56.3%

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

      4. distribute-rgt-neg-in 56.3%

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

      5. fma-define 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in ux around 0 99.2%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      2. expm1-undefine 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      3. mul-1-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{-ux \cdot {\left(maxCos - 1\right)}^{2}}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      4. sub-neg 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(maxCos + \left(-1\right)\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      5. metadata-eval 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(maxCos + \color{blue}{-1}\right)}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

      6. +-commutative 99.2%

         \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(e^{\mathsf{log1p}\left(-ux \cdot {\color{blue}{\left(-1 + maxCos\right)}}^{2}\right)} - 1\right) + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   7. Applied egg-rr 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(-ux \cdot {\left(-1 + maxCos\right)}^{2}\right)} - 1\right)} + -1 \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   8. Taylor expanded in maxCos around 0 98.4%

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

   if 4.99999987e-5 < maxCos

   1. Initial program 61.1%

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

      1. associate-*l* 61.1%

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

      2. sub-neg 61.1%

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

      3. +-commutative 61.1%

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

      4. distribute-rgt-neg-in 61.1%

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

      5. fma-define 61.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in uy around 0 52.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in ux around 0 80.8%

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

      1. associate--l+ 80.8%

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

      2. associate-*r* 80.8%

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

      3. neg-mul-1 80.8%

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

      4. sub-neg 80.8%

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

      5. metadata-eval 80.8%

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

      6. +-commutative 80.8%

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

   10. Simplified 80.8%

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

   11. Taylor expanded in maxCos around 0 80.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - \left(2 \cdot maxCos + ux \cdot \left(1 + maxCos \cdot \left(maxCos - 2\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in ux around 0 99.2%

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

6. Taylor expanded in maxCos around 0 97.7%

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

   1. neg-mul-1 97.7%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - 2 \cdot maxCos\right) - ux\right)} \]
10. Add Preprocessing

Alternative 8: 80.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - ((2.0e0 * maxcos) + (ux * (1.0e0 + (maxcos * (maxcos - 2.0e0))))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in uy around 0 49.4%

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

7. Simplified 49.5%

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

8. Taylor expanded in ux around 0 81.7%

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

   1. associate--l+ 81.7%

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

   2. associate-*r* 81.7%

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

   3. neg-mul-1 81.7%

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

   4. sub-neg 81.7%

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

   5. metadata-eval 81.7%

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

   6. +-commutative 81.7%

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

10. Simplified 81.7%

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

11. Taylor expanded in maxCos around 0 81.7%

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

12. Final simplification 81.7%

    \[\leadsto \sqrt{ux \cdot \left(2 - \left(2 \cdot maxCos + ux \cdot \left(1 + maxCos \cdot \left(maxCos - 2\right)\right)\right)\right)} \]
13. Add Preprocessing

Alternative 9: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f + (((2.0f * (ux * maxCos)) - ux) - (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 * (ux * maxcos)) - ux) - (2.0e0 * maxcos)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in uy around 0 49.4%

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

7. Simplified 49.5%

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

8. Taylor expanded in ux around 0 81.7%

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

   1. associate--l+ 81.7%

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

   2. associate-*r* 81.7%

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

   3. neg-mul-1 81.7%

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

   4. sub-neg 81.7%

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

   5. metadata-eval 81.7%

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

   6. +-commutative 81.7%

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

10. Simplified 81.7%

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

11. Taylor expanded in maxCos around 0 81.6%

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

    1. neg-mul-1 81.6%

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

    2. +-commutative 81.6%

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

    3. unsub-neg 81.6%

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

    4. *-commutative 81.6%

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

13. Simplified 81.6%

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

Alternative 10: 79.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f - (ux + (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 - (ux + (2.0e0 * maxcos)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in uy around 0 49.4%

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

7. Simplified 49.5%

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

8. Taylor expanded in ux around 0 81.7%

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

   1. associate--l+ 81.7%

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

   2. associate-*r* 81.7%

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

   3. neg-mul-1 81.7%

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

   4. sub-neg 81.7%

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

   5. metadata-eval 81.7%

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

   6. +-commutative 81.7%

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

10. Simplified 81.7%

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

11. Taylor expanded in maxCos around 0 81.0%

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

12. Final simplification 81.0%

    \[\leadsto \sqrt{ux \cdot \left(2 - \left(ux + 2 \cdot maxCos\right)\right)} \]
13. Add Preprocessing

Alternative 11: 75.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in uy around 0 49.4%

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

7. Simplified 49.5%

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

8. Taylor expanded in ux around 0 81.7%

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

   1. associate--l+ 81.7%

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

   2. associate-*r* 81.7%

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

   3. neg-mul-1 81.7%

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

   4. sub-neg 81.7%

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

   5. metadata-eval 81.7%

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

   6. +-commutative 81.7%

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

10. Simplified 81.7%

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

11. Taylor expanded in maxCos around 0 77.8%

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

    1. neg-mul-1 77.8%

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

    2. sub-neg 77.8%

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

13. Simplified 77.8%

    \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
14. Add Preprocessing

Alternative 12: 62.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

   1. associate-*l* 56.9%

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

   2. sub-neg 56.9%

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

   3. +-commutative 56.9%

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

   4. distribute-rgt-neg-in 56.9%

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

   5. fma-define 56.8%

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

3. Simplified 56.9%

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

5. Taylor expanded in uy around 0 49.4%

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

7. Simplified 49.5%

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

8. Taylor expanded in ux around 0 66.5%

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

9. Taylor expanded in maxCos around 0 64.2%

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

10. Final simplification 64.2%

    \[\leadsto \sqrt{2 \cdot ux} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024151 
(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)))))))