UniformSampleCone, x

Percentage Accurate: 56.9% → 98.9%

Time: 18.5s

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: 56.9% 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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((1.0f + ((1.0f - maxCos) + (ux * ((-1.0f + maxCos) * (1.0f - maxCos))))) - 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) + Float32(Float32(Float32(1.0) - maxCos) + Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))))) - maxCos))))
end

MATLAB

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

Derivation

1. Initial program 57.0%

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

   1. associate-*l* 57.0%

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

   2. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-define 57.0%

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

3. Simplified 57.1%

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

5. Taylor expanded in ux around inf 98.8%

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

6. Taylor expanded in ux around 0 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 92.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(uy * Float32(2.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(ux * Float32(Float32(maxCos * Float32(-2.0)) - Float32(ux * (Float32(Float32(-1.0) + maxCos) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos((single(pi) * (uy * single(2.0))));
	tmp = single(0.0);
	if (t_0 <= single(1.0))
		tmp = t_0 * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt(((single(2.0) * ux) + (ux * ((maxCos * single(-2.0)) - (ux * ((single(-1.0) + maxCos) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 1

   1. Initial program 57.0%

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

   3. Taylor expanded in ux around 0 99.0%

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

      1. associate--l+ 99.0%

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

      2. associate-*r* 99.0%

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

      3. mul-1-neg 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

      6. +-commutative 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in maxCos around 0 92.5%

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

      1. neg-mul-1 92.5%

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

      2. unsub-neg 92.5%

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

   8. Simplified 92.5%

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

   if 1 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.0%

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

   3. Taylor expanded in ux around 0 99.0%

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

      1. associate--l+ 99.0%

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

      2. associate-*r* 99.0%

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

      3. mul-1-neg 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

      6. +-commutative 99.0%

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

   5. Simplified 99.0%

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

      1. distribute-lft-in 99.1%

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

      2. cancel-sign-sub-inv 99.1%

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

      3. fma-define 99.1%

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

      4. metadata-eval 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in uy around 0 80.3%

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

4. Final simplification 92.5%

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

Alternative 3: 92.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(uy * Float32(2.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos((single(pi) * (uy * single(2.0))));
	tmp = single(0.0);
	if (t_0 <= single(1.0))
		tmp = t_0 * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt((ux * (single(2.0) + ((ux * ((single(-1.0) + maxCos) * (single(1.0) - maxCos))) - (single(2.0) * maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 1

   1. Initial program 57.0%

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

   3. Taylor expanded in ux around 0 99.0%

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

      1. associate--l+ 99.0%

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

      2. associate-*r* 99.0%

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

      3. mul-1-neg 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

      6. +-commutative 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in maxCos around 0 92.5%

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

      1. neg-mul-1 92.5%

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

      2. unsub-neg 92.5%

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

   8. Simplified 92.5%

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

   if 1 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. sub-neg 57.0%

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

      3. +-commutative 57.0%

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

      4. distribute-rgt-neg-in 57.0%

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

      5. fma-define 57.0%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in uy around 0 49.0%

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

      1. mul-1-neg 49.0%

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

      2. unsub-neg 49.0%

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

      3. sub-neg 49.0%

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

      4. metadata-eval 49.0%

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

      5. distribute-lft-in 49.0%

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

      6. *-commutative 49.0%

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

      7. mul-1-neg 49.0%

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

      8. sub-neg 49.0%

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

      9. *-commutative 49.0%

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

      10. associate--l+ 48.9%

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

      11. unpow2 48.9%

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

      12. sub-neg 48.9%

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

   7. Simplified 49.0%

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

   8. Taylor expanded in ux around -inf 80.0%

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

      1. sub-neg 80.0%

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

      2. mul-1-neg 80.0%

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

      3. +-commutative 80.0%

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

      4. unsub-neg 80.0%

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

      5. +-commutative 80.0%

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

      6. mul-1-neg 80.0%

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

      7. unsub-neg 80.0%

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

      8. sub-neg 80.0%

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

      9. metadata-eval 80.0%

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

      10. mul-1-neg 80.0%

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

      11. *-commutative 80.0%

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

      12. sub-neg 80.0%

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

      13. metadata-eval 80.0%

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

      14. +-commutative 80.0%

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

   10. Simplified 80.0%

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

   11. Taylor expanded in ux around 0 80.2%

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

       1. associate--l+ 80.2%

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

       2. *-commutative 80.2%

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

       3. sub-neg 80.2%

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

       4. metadata-eval 80.2%

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

       5. +-commutative 80.2%

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

       6. +-commutative 80.2%

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

   13. Simplified 80.2%

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

4. Final simplification 92.5%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

   1. associate-*l* 57.0%

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

   2. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-define 57.0%

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

3. Simplified 57.1%

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

5. Taylor expanded in ux around inf 98.8%

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

6. Taylor expanded in ux around 0 99.1%

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

   1. associate--l+ 99.1%

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

   2. +-commutative 99.1%

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

   3. mul-1-neg 99.1%

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

   4. unsub-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. sub-neg 99.1%

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

   7. metadata-eval 99.1%

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

   8. sub-neg 99.1%

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

   9. metadata-eval 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around 0 99.1%

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

7. Final simplification 99.1%

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

Alternative 6: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around 0 98.1%

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

   1. *-commutative 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 7: 89.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0013000000035390258)
   (sqrt
    (*
     ux
     (+ 2.0 (- (* ux (* (+ -1.0 maxCos) (- 1.0 maxCos))) (* 2.0 maxCos)))))
   (* (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((ux * (2.0f + ((ux * ((-1.0f + maxCos) * (1.0f - maxCos))) - (2.0f * maxCos)))));
	} 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(Float32(ux * Float32(Float32(2.0) + Float32(Float32(ux * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0013000000035390258))
		tmp = sqrt((ux * (single(2.0) + ((ux * ((single(-1.0) + maxCos) * (single(1.0) - maxCos))) - (single(2.0) * maxCos)))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0013

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. sub-neg 58.1%

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

      3. +-commutative 58.1%

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

      4. distribute-rgt-neg-in 58.1%

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

      5. fma-define 58.1%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in uy around 0 57.8%

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

      1. mul-1-neg 57.8%

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

      2. unsub-neg 57.8%

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

      3. sub-neg 57.8%

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

      4. metadata-eval 57.8%

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

      5. distribute-lft-in 57.8%

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

      6. *-commutative 57.8%

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

      7. mul-1-neg 57.8%

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

      8. sub-neg 57.8%

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

      9. *-commutative 57.8%

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

      10. associate--l+ 57.6%

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

      11. unpow2 57.6%

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

      12. sub-neg 57.6%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in ux around -inf 97.1%

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

      1. sub-neg 97.1%

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

      2. mul-1-neg 97.1%

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

      3. +-commutative 97.1%

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

      4. unsub-neg 97.1%

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

      5. +-commutative 97.1%

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

      6. mul-1-neg 97.1%

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

      7. unsub-neg 97.1%

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

      8. sub-neg 97.1%

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

      9. metadata-eval 97.1%

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

      10. mul-1-neg 97.1%

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

      11. *-commutative 97.1%

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

      12. sub-neg 97.1%

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

      13. metadata-eval 97.1%

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

      14. +-commutative 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in ux around 0 97.4%

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

       1. associate--l+ 97.4%

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

       2. *-commutative 97.4%

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

       3. sub-neg 97.4%

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

       4. metadata-eval 97.4%

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

       5. +-commutative 97.4%

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

       6. +-commutative 97.4%

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

   13. Simplified 97.4%

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

   if 0.0013 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.7%

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

      1. associate-*l* 54.7%

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

      2. sub-neg 54.7%

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

      3. +-commutative 54.7%

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

      4. distribute-rgt-neg-in 54.7%

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

      5. fma-define 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in maxCos around 0 51.4%

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

   6. Taylor expanded in ux around 0 73.4%

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

4. Final simplification 89.6%

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

Alternative 8: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around 0 96.7%

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

7. Final simplification 96.7%

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

Alternative 9: 80.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f + ((ux * ((-1.0f + maxCos) * (1.0f - maxCos))) - (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 * (((-1.0e0) + maxcos) * (1.0e0 - maxcos))) - (2.0e0 * maxcos)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

   1. associate-*l* 57.0%

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

   2. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-define 57.0%

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

3. Simplified 57.1%

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

5. Taylor expanded in uy around 0 49.0%

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

   1. mul-1-neg 49.0%

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

   2. unsub-neg 49.0%

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

   3. sub-neg 49.0%

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

   4. metadata-eval 49.0%

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

   5. distribute-lft-in 49.0%

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

   6. *-commutative 49.0%

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

   7. mul-1-neg 49.0%

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

   8. sub-neg 49.0%

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

   9. *-commutative 49.0%

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

   10. associate--l+ 48.9%

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

   11. unpow2 48.9%

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

   12. sub-neg 48.9%

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

7. Simplified 49.0%

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

8. Taylor expanded in ux around -inf 80.0%

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

   1. sub-neg 80.0%

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

   2. mul-1-neg 80.0%

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

   3. +-commutative 80.0%

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

   4. unsub-neg 80.0%

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

   5. +-commutative 80.0%

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

   6. mul-1-neg 80.0%

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

   7. unsub-neg 80.0%

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

   8. sub-neg 80.0%

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

   9. metadata-eval 80.0%

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

   10. mul-1-neg 80.0%

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

   11. *-commutative 80.0%

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

   12. sub-neg 80.0%

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

   13. metadata-eval 80.0%

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

   14. +-commutative 80.0%

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

10. Simplified 80.0%

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

11. Taylor expanded in ux around 0 80.2%

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

    1. associate--l+ 80.2%

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

    2. *-commutative 80.2%

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

    3. sub-neg 80.2%

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

    4. metadata-eval 80.2%

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

    5. +-commutative 80.2%

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

    6. +-commutative 80.2%

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

13. Simplified 80.2%

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

14. Final simplification 80.2%

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

Alternative 10: 64.8% accurate, 2.1× 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 57.0%

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

   1. associate-*l* 57.0%

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

   2. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-define 57.0%

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

3. Simplified 57.1%

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

5. Taylor expanded in uy around 0 49.0%

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

   1. mul-1-neg 49.0%

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

   2. unsub-neg 49.0%

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

   3. sub-neg 49.0%

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

   4. metadata-eval 49.0%

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

   5. distribute-lft-in 49.0%

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

   6. *-commutative 49.0%

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

   7. mul-1-neg 49.0%

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

   8. sub-neg 49.0%

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

   9. *-commutative 49.0%

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

   10. associate--l+ 48.9%

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

   11. unpow2 48.9%

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

   12. sub-neg 48.9%

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

7. Simplified 49.0%

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

8. Taylor expanded in ux around 0 64.7%

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

9. Final simplification 64.7%

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

Alternative 11: 75.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ ux \cdot \sqrt{-1 + \frac{2}{ux}} \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return ux * sqrtf((-1.0f + (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 = ux * sqrt(((-1.0e0) + (2.0e0 / ux)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

   1. associate-*l* 57.0%

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

   2. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-define 57.0%

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

3. Simplified 57.1%

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

5. Taylor expanded in uy around 0 49.0%

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

   1. mul-1-neg 49.0%

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

   2. unsub-neg 49.0%

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

   3. sub-neg 49.0%

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

   4. metadata-eval 49.0%

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

   5. distribute-lft-in 49.0%

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

   6. *-commutative 49.0%

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

   7. mul-1-neg 49.0%

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

   8. sub-neg 49.0%

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

   9. *-commutative 49.0%

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

   10. associate--l+ 48.9%

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

   11. unpow2 48.9%

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

   12. sub-neg 48.9%

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

7. Simplified 49.0%

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

8. Taylor expanded in ux around -inf 80.0%

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

   1. sub-neg 80.0%

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

   2. mul-1-neg 80.0%

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

   3. +-commutative 80.0%

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

   4. unsub-neg 80.0%

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

   5. +-commutative 80.0%

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

   6. mul-1-neg 80.0%

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

   7. unsub-neg 80.0%

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

   8. sub-neg 80.0%

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

   9. metadata-eval 80.0%

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

   10. mul-1-neg 80.0%

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

   11. *-commutative 80.0%

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

   12. sub-neg 80.0%

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

   13. metadata-eval 80.0%

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

   14. +-commutative 80.0%

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

10. Simplified 80.0%

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

11. Taylor expanded in maxCos around 0 75.5%

    \[\leadsto \color{blue}{ux \cdot \sqrt{2 \cdot \frac{1}{ux} - 1}} \]
12. Step-by-step derivation

    1. sub-neg 75.5%

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

    2. associate-*r/ 75.5%

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

    3. metadata-eval 75.5%

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

    4. metadata-eval 75.5%

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

13. Simplified 75.5%

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

14. Final simplification 75.5%

    \[\leadsto ux \cdot \sqrt{-1 + \frac{2}{ux}} \]
15. Add Preprocessing

Alternative 12: 62.0% 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 57.0%

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

   1. associate-*l* 57.0%

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

   2. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-define 57.0%

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

3. Simplified 57.1%

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

5. Taylor expanded in uy around 0 49.0%

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

   1. mul-1-neg 49.0%

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

   2. unsub-neg 49.0%

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

   3. sub-neg 49.0%

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

   4. metadata-eval 49.0%

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

   5. distribute-lft-in 49.0%

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

   6. *-commutative 49.0%

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

   7. mul-1-neg 49.0%

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

   8. sub-neg 49.0%

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

   9. *-commutative 49.0%

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

   10. associate--l+ 48.9%

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

   11. unpow2 48.9%

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

   12. sub-neg 48.9%

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

7. Simplified 49.0%

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

8. Taylor expanded in ux around 0 64.7%

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

9. Taylor expanded in maxCos around 0 62.2%

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

10. Final simplification 62.2%

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

Reproduce

herbie shell --seed 2024059 
(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)))))))