UniformSampleCone, x

Percentage Accurate: 57.3% → 99.0%

Time: 18.1s

Alternatives: 21

Speedup: 3.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.3% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.1%

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

   1. add-log-exp 99.2%

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

   2. +-commutative 99.2%

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

   3. fma-def 99.2%

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

   4. sub-neg 99.2%

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

   5. metadata-eval 99.2%

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

   6. +-commutative 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.1%

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

   1. fma-def 99.1%

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

   2. associate--l+ 99.2%

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

   3. mul-1-neg 99.2%

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

   4. sub-neg 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-neg-in 99.2%

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

   7. metadata-eval 99.2%

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

   8. +-commutative 99.2%

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

   9. sub-neg 99.2%

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

   10. *-commutative 99.2%

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

   11. sub-neg 99.2%

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

   12. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in uy around inf 99.1%

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

   1. *-commutative 99.1%

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

   2. fma-def 99.2%

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

   3. sub-neg 99.2%

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

   4. mul-1-neg 99.2%

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

   5. mul-1-neg 99.2%

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

   6. sub-neg 99.2%

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

   7. sub-neg 99.2%

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

   8. metadata-eval 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 90.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999449849128723:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* uy 2.0)))))
   (if (<= t_0 0.9999449849128723)
     (* t_0 (sqrt (* ux (- 2.0 (* maxCos 2.0)))))
     (sqrt
      (+
       (* (pow ux 2.0) (* (+ -1.0 maxCos) (- 1.0 maxCos)))
       (* ux (- (+ 1.0 (- 1.0 maxCos)) 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 <= 0.9999449849128723f) {
		tmp = t_0 * sqrtf((ux * (2.0f - (maxCos * 2.0f))));
	} else {
		tmp = sqrtf(((powf(ux, 2.0f) * ((-1.0f + maxCos) * (1.0f - maxCos))) + (ux * ((1.0f + (1.0f - maxCos)) - 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(0.9999449849128723))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * Float32(2.0))))));
	else
		tmp = sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))) + Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - 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(0.9999449849128723))
		tmp = t_0 * sqrt((ux * (single(2.0) - (maxCos * single(2.0)))));
	else
		tmp = sqrt((((ux ^ single(2.0)) * ((single(-1.0) + maxCos) * (single(1.0) - maxCos))) + (ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) < 0.999944985

   1. Initial program 60.3%

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

   2. Taylor expanded in ux around 0 73.9%

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

   if 0.999944985 < (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32)))

   1. Initial program 53.9%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in ux around 0 99.6%

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

   5. Taylor expanded in uy around 0 97.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999449849128723:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \end{array} \]

Alternative 4: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999449849128723:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* uy 2.0)))))
   (if (<= t_0 0.9999449849128723)
     (* t_0 (sqrt (* ux 2.0)))
     (sqrt
      (+
       (* (pow ux 2.0) (* (+ -1.0 maxCos) (- 1.0 maxCos)))
       (* ux (- (+ 1.0 (- 1.0 maxCos)) 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 <= 0.9999449849128723f) {
		tmp = t_0 * sqrtf((ux * 2.0f));
	} else {
		tmp = sqrtf(((powf(ux, 2.0f) * ((-1.0f + maxCos) * (1.0f - maxCos))) + (ux * ((1.0f + (1.0f - maxCos)) - 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(0.9999449849128723))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(2.0))));
	else
		tmp = sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(1.0) - maxCos))) + Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - 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(0.9999449849128723))
		tmp = t_0 * sqrt((ux * single(2.0)));
	else
		tmp = sqrt((((ux ^ single(2.0)) * ((single(-1.0) + maxCos) * (single(1.0) - maxCos))) + (ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) < 0.999944985

   1. Initial program 60.3%

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

   2. Taylor expanded in ux around 0 45.0%

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

   3. Taylor expanded in maxCos around 0 71.3%

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

      1. *-commutative 71.3%

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

   5. Simplified 71.3%

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

   if 0.999944985 < (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32)))

   1. Initial program 53.9%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in ux around 0 99.6%

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

   5. Taylor expanded in uy around 0 97.2%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999449849128723:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(-1 + maxCos\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.1%

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

5. Final simplification 99.1%

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

Alternative 6: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around -inf 99.1%

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

   1. +-commutative 99.1%

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

   2. mul-1-neg 99.1%

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

   3. unsub-neg 99.1%

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

   4. *-commutative 99.1%

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

   5. mul-1-neg 99.1%

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

   6. sub-neg 99.1%

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

   7. sub-neg 99.1%

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

   8. metadata-eval 99.1%

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

   9. sub-neg 99.1%

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

   10. metadata-eval 99.1%

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

   11. +-commutative 99.1%

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

   12. mul-1-neg 99.1%

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

   13. unsub-neg 99.1%

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

   14. mul-1-neg 99.1%

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

   15. sub-neg 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in uy around inf 99.1%

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

8. Final simplification 99.1%

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

Alternative 7: 97.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.1%

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

5. Taylor expanded in maxCos around 0 97.5%

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

6. Final simplification 97.5%

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

Alternative 8: 92.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.1%

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

   1. fma-def 99.1%

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

   2. associate--l+ 99.2%

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

   3. mul-1-neg 99.2%

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

   4. sub-neg 99.2%

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

   5. metadata-eval 99.2%

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

   6. distribute-neg-in 99.2%

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

   7. metadata-eval 99.2%

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

   8. +-commutative 99.2%

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

   9. sub-neg 99.2%

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

   10. *-commutative 99.2%

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

   11. sub-neg 99.2%

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

   12. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in maxCos around 0 94.8%

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

   1. +-commutative 94.8%

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

   2. mul-1-neg 94.8%

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

   3. unsub-neg 94.8%

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

   4. *-commutative 94.8%

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

9. Simplified 94.8%

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

10. Final simplification 94.8%

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

Alternative 9: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)\\ \mathbf{if}\;t_0 \leq 0.00022000000171829015:\\ \;\;\;\;\cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (+
          1.0
          (* (+ (- 1.0 ux) (* ux maxCos)) (- (+ ux -1.0) (* ux maxCos))))))
   (if (<= t_0 0.00022000000171829015)
     (*
      (cos (* (* PI -2.0) uy))
      (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos))))
     (* (cos (* PI (* uy 2.0))) (sqrt t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 1.0f + (((1.0f - ux) + (ux * maxCos)) * ((ux + -1.0f) - (ux * maxCos)));
	float tmp;
	if (t_0 <= 0.00022000000171829015f) {
		tmp = cosf(((((float) M_PI) * -2.0f) * uy)) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) * Float32(Float32(ux + Float32(-1.0)) - Float32(ux * maxCos))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00022000000171829015))
		tmp = Float32(cos(Float32(Float32(Float32(pi) * Float32(-2.0)) * uy)) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(1.0) + (((single(1.0) - ux) + (ux * maxCos)) * ((ux + single(-1.0)) - (ux * maxCos)));
	tmp = single(0.0);
	if (t_0 <= single(0.00022000000171829015))
		tmp = cos(((single(pi) * single(-2.0)) * uy)) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 (*.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)))) < 2.20000002e-4

   1. Initial program 32.6%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in ux around 0 94.3%

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

   if 2.20000002e-4 < (-.f32 1 (*.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))))

   1. Initial program 89.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right) \leq 0.00022000000171829015:\\ \;\;\;\;\cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \end{array} \]

Alternative 10: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right) \leq 0.00022000000171829015:\\ \;\;\;\;\cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<=
      (+ 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (- (+ ux -1.0) (* ux maxCos))))
      0.00022000000171829015)
   (* (cos (* (* PI -2.0) uy)) (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos))))
   (*
    (cos (* -2.0 (* PI uy)))
    (sqrt
     (-
      1.0
      (* (+ 1.0 (* ux (+ -1.0 maxCos))) (- (+ 1.0 (* ux maxCos)) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((1.0f + (((1.0f - ux) + (ux * maxCos)) * ((ux + -1.0f) - (ux * maxCos)))) <= 0.00022000000171829015f) {
		tmp = cosf(((((float) M_PI) * -2.0f) * uy)) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = cosf((-2.0f * (((float) M_PI) * uy))) * sqrtf((1.0f - ((1.0f + (ux * (-1.0f + maxCos))) * ((1.0f + (ux * maxCos)) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) * Float32(Float32(ux + Float32(-1.0)) - Float32(ux * maxCos)))) <= Float32(0.00022000000171829015))
		tmp = Float32(cos(Float32(Float32(Float32(pi) * Float32(-2.0)) * uy)) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))));
	else
		tmp = Float32(cos(Float32(Float32(-2.0) * Float32(Float32(pi) * uy))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(Float32(Float32(1.0) + Float32(ux * maxCos)) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(1.0) + (((single(1.0) - ux) + (ux * maxCos)) * ((ux + single(-1.0)) - (ux * maxCos)))) <= single(0.00022000000171829015))
		tmp = cos(((single(pi) * single(-2.0)) * uy)) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = cos((single(-2.0) * (single(pi) * uy))) * sqrt((single(1.0) - ((single(1.0) + (ux * (single(-1.0) + maxCos))) * ((single(1.0) + (ux * maxCos)) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 (*.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)))) < 2.20000002e-4

   1. Initial program 32.6%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in ux around 0 94.3%

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

   if 2.20000002e-4 < (-.f32 1 (*.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))))

   1. Initial program 89.5%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in uy around inf 89.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right) \leq 0.00022000000171829015:\\ \;\;\;\;\cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\\ \end{array} \]

Alternative 11: 79.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.1%

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

5. Taylor expanded in uy around 0 78.8%

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

6. Final simplification 78.8%

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

Alternative 12: 79.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around -inf 99.1%

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

   1. +-commutative 99.1%

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

   2. mul-1-neg 99.1%

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

   3. unsub-neg 99.1%

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

   4. *-commutative 99.1%

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

   5. mul-1-neg 99.1%

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

   6. sub-neg 99.1%

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

   7. sub-neg 99.1%

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

   8. metadata-eval 99.1%

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

   9. sub-neg 99.1%

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

   10. metadata-eval 99.1%

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

   11. +-commutative 99.1%

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

   12. mul-1-neg 99.1%

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

   13. unsub-neg 99.1%

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

   14. mul-1-neg 99.1%

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

   15. sub-neg 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in uy around 0 78.7%

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

8. Final simplification 78.7%

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

Alternative 13: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-1, -1 + maxCos, 1\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (* ux (- (fma -1.0 (+ -1.0 maxCos) 1.0) maxCos)))
   (sqrt
    (- 1.0 (* (+ 1.0 (* ux (+ -1.0 maxCos))) (- (+ 1.0 (* ux maxCos)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((ux * (fmaf(-1.0f, (-1.0f + maxCos), 1.0f) - maxCos)));
	} else {
		tmp = sqrtf((1.0f - ((1.0f + (ux * (-1.0f + maxCos))) * ((1.0f + (ux * maxCos)) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(ux * Float32(fma(Float32(-1.0), Float32(Float32(-1.0) + maxCos), Float32(1.0)) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(Float32(Float32(1.0) + Float32(ux * maxCos)) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 32.4%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in uy around 0 29.5%

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

      1. add-exp-log 29.5%

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

      2. pow1/2 29.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(1 + -1 \cdot \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)}^{0.5}\right)}} \]

      3. log-pow 29.5%

         \[\leadsto e^{\color{blue}{0.5 \cdot \log \left(1 + -1 \cdot \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)}} \]

      4. log1p-udef 29.5%

         \[\leadsto e^{0.5 \cdot \color{blue}{\mathsf{log1p}\left(-1 \cdot \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)}} \]

      5. mul-1-neg 29.5%

         \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(\color{blue}{-\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)} \]

      6. +-commutative 29.5%

         \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(-\color{blue}{\left(ux \cdot \left(maxCos - 1\right) + 1\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)} \]

      7. fma-def 29.5%

         \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(-\color{blue}{\mathsf{fma}\left(ux, maxCos - 1, 1\right)} \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)} \]

      8. sub-neg 29.5%

         \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(-\mathsf{fma}\left(ux, \color{blue}{maxCos + \left(-1\right)}, 1\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)} \]

      9. metadata-eval 29.5%

         \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(-\mathsf{fma}\left(ux, maxCos + \color{blue}{-1}, 1\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)} \]

      10. +-commutative 29.5%

          \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(-\mathsf{fma}\left(ux, maxCos + -1, 1\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux + 1\right)} - ux\right)\right)} \]

      11. *-commutative 29.5%

          \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(-\mathsf{fma}\left(ux, maxCos + -1, 1\right) \cdot \left(\left(\color{blue}{ux \cdot maxCos} + 1\right) - ux\right)\right)} \]

      12. fma-def 29.5%

          \[\leadsto e^{0.5 \cdot \mathsf{log1p}\left(-\mathsf{fma}\left(ux, maxCos + -1, 1\right) \cdot \left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)} - ux\right)\right)} \]

   6. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{e^{0.5 \cdot \mathsf{log1p}\left(-\mathsf{fma}\left(ux, maxCos + -1, 1\right) \cdot \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)\right)}} \]
   7. Step-by-step derivation

      1. exp-prod 29.5%

         \[\leadsto \color{blue}{{\left(e^{0.5}\right)}^{\left(\mathsf{log1p}\left(-\mathsf{fma}\left(ux, maxCos + -1, 1\right) \cdot \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)\right)\right)}} \]

      2. distribute-rgt-neg-in 29.5%

         \[\leadsto {\left(e^{0.5}\right)}^{\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(ux, maxCos + -1, 1\right) \cdot \left(-\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)\right)}\right)\right)} \]

   8. Simplified 29.5%

      \[\leadsto \color{blue}{{\left(e^{0.5}\right)}^{\left(\mathsf{log1p}\left(\mathsf{fma}\left(ux, maxCos + -1, 1\right) \cdot \left(-\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)\right)\right)\right)}} \]

   9. Taylor expanded in ux around 0 77.3%

      \[\leadsto \color{blue}{e^{0.5 \cdot \left(\log ux + \log \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 77.3%

          \[\leadsto e^{\color{blue}{\left(\log ux + \log \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right) \cdot 0.5}} \]

       2. sum-log 77.7%

          \[\leadsto e^{\color{blue}{\log \left(ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)} \cdot 0.5} \]

       3. exp-to-pow 79.4%

          \[\leadsto \color{blue}{{\left(ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)\right)}^{0.5}} \]

       4. pow1/2 79.4%

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

       5. +-commutative 79.4%

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

       6. fma-def 79.4%

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

       7. sub-neg 79.4%

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

       8. metadata-eval 79.4%

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

       9. +-commutative 79.4%

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

   11. Applied egg-rr 79.4%

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

   if 1.10000001e-4 < ux

   1. Initial program 89.2%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in uy around 0 67.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-1, -1 + maxCos, 1\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\\ \end{array} \]

Alternative 14: 74.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{\left(--2 \cdot ux\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(1 + \left(ux \cdot maxCos - ux\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (- (- (* -2.0 ux)) (* 2.0 (* ux maxCos))))
   (sqrt
    (- 1.0 (* (+ 1.0 (* ux (+ -1.0 maxCos))) (+ 1.0 (- (* ux maxCos) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((-(-2.0f * ux) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f - ((1.0f + (ux * (-1.0f + maxCos))) * (1.0f + ((ux * maxCos) - ux)))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((-((-2.0e0) * ux) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 + (ux * ((-1.0e0) + maxcos))) * (1.0e0 + ((ux * maxcos) - ux)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(Float32(-Float32(Float32(-2.0) * ux)) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(Float32(1.0) + Float32(Float32(ux * maxCos) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((-(single(-2.0) * ux) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) + (ux * (single(-1.0) + maxCos))) * (single(1.0) + ((ux * maxCos) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 32.4%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in uy around 0 29.5%

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

   5. Taylor expanded in ux around 0 79.4%

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

   6. Taylor expanded in maxCos around 0 79.4%

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

   if 1.10000001e-4 < ux

   1. Initial program 89.2%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in uy around 0 67.8%

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

      1. associate--l+ 67.7%

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

      2. *-commutative 67.7%

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

   6. Applied egg-rr 67.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{\left(--2 \cdot ux\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(1 + \left(ux \cdot maxCos - ux\right)\right)}\\ \end{array} \]

Alternative 15: 74.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{\left(--2 \cdot ux\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (- (- (* -2.0 ux)) (* 2.0 (* ux maxCos))))
   (sqrt
    (- 1.0 (* (+ 1.0 (* ux (+ -1.0 maxCos))) (- (+ 1.0 (* ux maxCos)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((-(-2.0f * ux) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f - ((1.0f + (ux * (-1.0f + maxCos))) * ((1.0f + (ux * maxCos)) - ux))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((-((-2.0e0) * ux) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 + (ux * ((-1.0e0) + maxcos))) * ((1.0e0 + (ux * maxcos)) - ux))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(Float32(-Float32(Float32(-2.0) * ux)) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(Float32(Float32(1.0) + Float32(ux * maxCos)) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((-(single(-2.0) * ux) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) + (ux * (single(-1.0) + maxCos))) * ((single(1.0) + (ux * maxCos)) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 32.4%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in uy around 0 29.5%

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

   5. Taylor expanded in ux around 0 79.4%

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

   6. Taylor expanded in maxCos around 0 79.4%

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

   if 1.10000001e-4 < ux

   1. Initial program 89.2%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in uy around 0 67.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{\left(--2 \cdot ux\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(\left(1 + ux \cdot maxCos\right) - ux\right)}\\ \end{array} \]

Alternative 16: 73.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{\left(--2 \cdot ux\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 + ux \cdot \left(-1 + maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (- (- (* -2.0 ux)) (* 2.0 (* ux maxCos))))
   (sqrt (- 1.0 (* (- 1.0 ux) (+ 1.0 (* ux (+ -1.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((-(-2.0f * ux) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f - ((1.0f - ux) * (1.0f + (ux * (-1.0f + maxCos))))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((-((-2.0e0) * ux) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 - ux) * (1.0e0 + (ux * ((-1.0e0) + maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(Float32(-Float32(Float32(-2.0) * ux)) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((-(single(-2.0) * ux) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) + (ux * (single(-1.0) + maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 32.4%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in uy around 0 29.5%

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

   5. Taylor expanded in ux around 0 79.4%

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

   6. Taylor expanded in maxCos around 0 79.4%

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

   if 1.10000001e-4 < ux

   1. Initial program 89.2%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in uy around 0 67.8%

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

   5. Taylor expanded in maxCos around 0 65.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{\left(--2 \cdot ux\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 + ux \cdot \left(-1 + maxCos\right)\right)}\\ \end{array} \]

Alternative 17: 73.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{\left(--2 \cdot ux\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (- (- (* -2.0 ux)) (* 2.0 (* ux maxCos))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((-(-2.0f * ux) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((-((-2.0e0) * ux) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(Float32(-Float32(Float32(-2.0) * ux)) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((-(single(-2.0) * ux) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 32.4%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in uy around 0 29.5%

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

   5. Taylor expanded in ux around 0 79.4%

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

   6. Taylor expanded in maxCos around 0 79.4%

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

   if 1.10000001e-4 < ux

   1. Initial program 89.2%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in uy around 0 67.8%

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

   5. Taylor expanded in maxCos around 0 65.1%

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

      1. mul-1-neg 65.1%

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

      2. unsub-neg 65.1%

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

      3. neg-mul-1 65.1%

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

      4. sub-neg 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{\left(--2 \cdot ux\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \]

Alternative 18: 73.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (sqrt (* ux (- 2.0 (* maxCos 2.0))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((ux * (2.0f - (maxCos * 2.0f))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt((ux * (2.0e0 - (maxcos * 2.0e0))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * Float32(2.0)))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt((ux * (single(2.0) - (maxCos * single(2.0)))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 32.4%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in uy around 0 29.5%

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

   5. Taylor expanded in ux around 0 79.4%

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

   if 1.10000001e-4 < ux

   1. Initial program 89.2%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in uy around 0 67.8%

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

   5. Taylor expanded in maxCos around 0 65.1%

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

      1. mul-1-neg 65.1%

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

      2. unsub-neg 65.1%

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

      3. neg-mul-1 65.1%

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

      4. sub-neg 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \]

Alternative 19: 64.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in uy around 0 45.2%

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

5. Taylor expanded in ux around 0 64.4%

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

6. Final simplification 64.4%

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

Alternative 20: 62.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

2. Taylor expanded in ux around 0 42.3%

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

3. Taylor expanded in maxCos around 0 75.1%

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

   1. *-commutative 75.1%

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

5. Simplified 75.1%

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

   1. add-sqr-sqrt 69.8%

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

   2. pow2 69.8%

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

   3. associate-*l* 69.8%

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

7. Applied egg-rr 69.8%

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

8. Taylor expanded in uy around 0 62.9%

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

9. Final simplification 62.9%

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

Alternative 21: 6.6% accurate, 3.2× 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 55.7%

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

3. Simplified 55.9%

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

4. Taylor expanded in uy around 0 45.2%

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

5. Taylor expanded in ux around 0 6.6%

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

6. Final simplification 6.6%

   \[\leadsto \sqrt{0} \]

Reproduce

herbie shell --seed 2023336 
(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)))))))