UniformSampleCone, x

Percentage Accurate: 57.3% → 99.0%

Time: 13.0s

Alternatives: 14

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

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

   7. distribute-lft-in 99.1%

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

   8. cancel-sign-sub-inv 99.1%

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

   9. fma-define 99.1%

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

   10. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.9999979734420776:\\ \;\;\;\;t\_0 \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + ux \cdot \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (* uy 2.0) PI))))
   (if (<= t_0 0.9999979734420776)
     (* t_0 (sqrt (- (* 2.0 ux) (* ux ux))))
     (sqrt
      (+
       (* 2.0 ux)
       (* ux (- (* maxCos (- (- (* 2.0 ux) (* ux maxCos)) 2.0)) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (t_0 <= 0.9999979734420776f) {
		tmp = t_0 * sqrtf(((2.0f * ux) - (ux * ux)));
	} else {
		tmp = sqrtf(((2.0f * ux) + (ux * ((maxCos * (((2.0f * ux) - (ux * maxCos)) - 2.0f)) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999979734420776))
		tmp = Float32(t_0 * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	else
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(ux * Float32(Float32(maxCos * Float32(Float32(Float32(Float32(2.0) * ux) - Float32(ux * maxCos)) - Float32(2.0))) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos(((uy * single(2.0)) * single(pi)));
	tmp = single(0.0);
	if (t_0 <= single(0.9999979734420776))
		tmp = t_0 * sqrt(((single(2.0) * ux) - (ux * ux)));
	else
		tmp = sqrt(((single(2.0) * ux) + (ux * ((maxCos * (((single(2.0) * ux) - (ux * maxCos)) - single(2.0))) - ux))));
	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))) < 0.999997973

   1. Initial program 59.8%

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

   3. Taylor expanded in ux around 0 98.5%

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

   4. Taylor expanded in maxCos around 0 91.8%

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

      1. neg-mul-1 91.8%

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

   6. Simplified 91.8%

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

      1. distribute-rgt-in 91.8%

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

      2. *-commutative 91.8%

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

   8. Applied egg-rr 91.8%

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

   if 0.999997973 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   3. Taylor expanded in ux around 0 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-*r* 99.3%

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

      3. neg-mul-1 99.3%

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

      4. sub-neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. +-commutative 99.3%

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

      7. distribute-lft-in 99.4%

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

      8. cancel-sign-sub-inv 99.4%

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

      9. fma-define 99.4%

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

      10. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in uy around 0 98.7%

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

   7. Taylor expanded in maxCos around 0 98.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999979734420776:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + ux \cdot \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999979734420776:\\ \;\;\;\;ux \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + ux \cdot \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.9999979734420776)
   (* ux (* (cos (* 2.0 (* uy PI))) (sqrt (+ -1.0 (/ 2.0 ux)))))
   (sqrt
    (+
     (* 2.0 ux)
     (* ux (- (* maxCos (- (- (* 2.0 ux) (* ux maxCos)) 2.0)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.9999979734420776f) {
		tmp = ux * (cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((-1.0f + (2.0f / ux))));
	} else {
		tmp = sqrtf(((2.0f * ux) + (ux * ((maxCos * (((2.0f * ux) - (ux * maxCos)) - 2.0f)) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.9999979734420776))
		tmp = Float32(ux * Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	else
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(ux * Float32(Float32(maxCos * Float32(Float32(Float32(Float32(2.0) * ux) - Float32(ux * maxCos)) - Float32(2.0))) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos(((uy * single(2.0)) * single(pi))) <= single(0.9999979734420776))
		tmp = ux * (cos((single(2.0) * (uy * single(pi)))) * sqrt((single(-1.0) + (single(2.0) / ux))));
	else
		tmp = sqrt(((single(2.0) * ux) + (ux * ((maxCos * (((single(2.0) * ux) - (ux * maxCos)) - single(2.0))) - ux))));
	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))) < 0.999997973

   1. Initial program 59.8%

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

   3. Taylor expanded in ux around -inf 98.2%

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

      1. +-commutative 98.2%

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

      2. metadata-eval 98.2%

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

      3. cancel-sign-sub-inv 98.2%

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

      4. associate-*r/ 98.2%

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

      5. metadata-eval 98.2%

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

      6. associate-*r/ 98.2%

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

      7. div-sub 98.4%

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

      8. cancel-sign-sub-inv 98.4%

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

      9. metadata-eval 98.4%

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

      10. +-commutative 98.4%

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

      11. metadata-eval 98.4%

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

      12. distribute-lft-neg-in 98.4%

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

      13. metadata-eval 98.4%

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

      14. distribute-neg-in 98.4%

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

      15. metadata-eval 98.4%

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

      16. sub-neg 98.4%

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

      17. fma-neg 98.4%

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

      18. metadata-eval 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in maxCos around 0 91.5%

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

      1. associate-*l* 91.8%

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

      2. sub-neg 91.8%

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

      3. associate-*r/ 91.8%

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

      4. metadata-eval 91.8%

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

      5. metadata-eval 91.8%

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

   8. Simplified 91.8%

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

   if 0.999997973 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   3. Taylor expanded in ux around 0 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-*r* 99.3%

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

      3. neg-mul-1 99.3%

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

      4. sub-neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. +-commutative 99.3%

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

      7. distribute-lft-in 99.4%

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

      8. cancel-sign-sub-inv 99.4%

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

      9. fma-define 99.4%

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

      10. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in uy around 0 98.7%

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

   7. Taylor expanded in maxCos around 0 98.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999979734420776:\\ \;\;\;\;ux \cdot \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + ux \cdot \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999979734420776:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + ux \cdot \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.9999979734420776)
   (* (cos (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 ux))))
   (sqrt
    (+
     (* 2.0 ux)
     (* ux (- (* maxCos (- (- (* 2.0 ux) (* ux maxCos)) 2.0)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.9999979734420776f) {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf(((2.0f * ux) + (ux * ((maxCos * (((2.0f * ux) - (ux * maxCos)) - 2.0f)) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.9999979734420776))
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(ux * Float32(Float32(maxCos * Float32(Float32(Float32(Float32(2.0) * ux) - Float32(ux * maxCos)) - Float32(2.0))) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos(((uy * single(2.0)) * single(pi))) <= single(0.9999979734420776))
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt(((single(2.0) * ux) + (ux * ((maxCos * (((single(2.0) * ux) - (ux * maxCos)) - single(2.0))) - ux))));
	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))) < 0.999997973

   1. Initial program 59.8%

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

   3. Taylor expanded in ux around 0 98.5%

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

   4. Taylor expanded in maxCos around 0 91.8%

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

      1. neg-mul-1 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in uy around inf 91.8%

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

   if 0.999997973 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   3. Taylor expanded in ux around 0 99.3%

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

      1. associate--l+ 99.3%

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

      2. associate-*r* 99.3%

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

      3. neg-mul-1 99.3%

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

      4. sub-neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. +-commutative 99.3%

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

      7. distribute-lft-in 99.4%

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

      8. cancel-sign-sub-inv 99.4%

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

      9. fma-define 99.4%

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

      10. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in uy around 0 98.7%

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

   7. Taylor expanded in maxCos around 0 98.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999979734420776:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + ux \cdot \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. Final simplification 99.0%

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

Alternative 6: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999600052833557:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + ux \cdot \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.9999600052833557)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))
   (sqrt
    (+
     (* 2.0 ux)
     (* ux (- (* maxCos (- (- (* 2.0 ux) (* ux maxCos)) 2.0)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.9999600052833557f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	} else {
		tmp = sqrtf(((2.0f * ux) + (ux * ((maxCos * (((2.0f * ux) - (ux * maxCos)) - 2.0f)) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.9999600052833557))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	else
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(ux * Float32(Float32(maxCos * Float32(Float32(Float32(Float32(2.0) * ux) - Float32(ux * maxCos)) - Float32(2.0))) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos(((uy * single(2.0)) * single(pi))) <= single(0.9999600052833557))
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	else
		tmp = sqrt(((single(2.0) * ux) + (ux * ((maxCos * (((single(2.0) * ux) - (ux * maxCos)) - single(2.0))) - ux))));
	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))) < 0.999960005

   1. Initial program 55.8%

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

      1. associate-*l* 55.8%

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

      2. sub-neg 55.8%

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

      3. +-commutative 55.8%

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

      4. distribute-rgt-neg-in 55.8%

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

      5. fma-define 55.5%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in maxCos around 0 55.0%

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

   6. Taylor expanded in ux around 0 73.9%

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

      1. *-commutative 73.9%

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

   8. Simplified 73.9%

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

   if 0.999960005 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 59.7%

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

   3. Taylor expanded in ux around 0 99.2%

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

      1. associate--l+ 99.2%

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

      2. associate-*r* 99.2%

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

      3. neg-mul-1 99.2%

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

      4. sub-neg 99.2%

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

      5. metadata-eval 99.2%

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

      6. +-commutative 99.2%

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

      7. distribute-lft-in 99.3%

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

      8. cancel-sign-sub-inv 99.3%

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

      9. fma-define 99.3%

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

      10. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in uy around 0 96.2%

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

   7. Taylor expanded in maxCos around 0 96.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999600052833557:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux + ux \cdot \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. Taylor expanded in maxCos around 0 99.0%

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

5. Taylor expanded in ux around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. unsub-neg 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 8: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. 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. Taylor expanded in maxCos around 0 98.2%

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

5. Final simplification 98.2%

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

Alternative 9: 79.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

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

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

   7. distribute-lft-in 99.1%

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

   8. cancel-sign-sub-inv 99.1%

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

   9. fma-define 99.1%

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

   10. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in uy around 0 79.0%

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

7. Taylor expanded in maxCos around 0 79.0%

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

8. Final simplification 79.0%

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

Alternative 10: 79.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

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

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

   7. distribute-lft-in 99.1%

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

   8. cancel-sign-sub-inv 99.1%

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

   9. fma-define 99.1%

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

   10. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in uy around 0 79.0%

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

7. Taylor expanded in maxCos around 0 79.0%

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

   1. neg-mul-1 79.0%

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

   2. +-commutative 79.0%

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

   3. unsub-neg 79.0%

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

   4. neg-mul-1 79.0%

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

   5. +-commutative 79.0%

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

   6. distribute-lft-neg-in 79.0%

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

   7. mul-1-neg 79.0%

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

   8. distribute-rgt-in 79.0%

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

   9. mul-1-neg 79.0%

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

   10. unsub-neg 79.0%

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

9. Simplified 79.0%

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

10. Final simplification 79.0%

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

Alternative 11: 78.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

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

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

   7. distribute-lft-in 99.1%

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

   8. cancel-sign-sub-inv 99.1%

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

   9. fma-define 99.1%

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

   10. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in uy around 0 79.0%

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

7. Taylor expanded in maxCos around 0 78.5%

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

8. Final simplification 78.5%

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

Alternative 12: 75.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.5%

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

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

   7. distribute-lft-in 99.1%

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

   8. cancel-sign-sub-inv 99.1%

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

   9. fma-define 99.1%

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

   10. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in uy around 0 79.0%

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

7. Taylor expanded in maxCos around 0 73.7%

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

   1. neg-mul-1 73.7%

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

9. Simplified 73.7%

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

10. Final simplification 73.7%

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

Alternative 13: 64.4% 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 58.5%

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

3. Taylor expanded in uy around 0 49.5%

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

4. Taylor expanded in ux around 0 63.9%

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

Alternative 14: 6.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt 0.0))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(0.0f);
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(0.0e0)
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(0.0))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt(single(0.0));
end

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in uy around 0 49.5%

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

4. Taylor expanded in ux around 0 6.6%

   \[\leadsto \sqrt{1 - \color{blue}{1}} \]

5. Final simplification 6.6%

   \[\leadsto \sqrt{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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)))))))