UniformSampleCone, x

Percentage Accurate: 57.6% → 98.9%

Time: 12.8s

Alternatives: 15

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in ux around inf 98.9%

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

6. Taylor expanded in ux around 0 99.0%

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

   1. associate-+r+ 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. +-commutative 99.0%

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

   5. distribute-lft-in 99.0%

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

   6. metadata-eval 99.0%

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

   7. mul-1-neg 99.0%

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

   8. sub-neg 99.0%

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

   9. *-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. metadata-eval 99.0%

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

   12. +-commutative 99.0%

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

8. Simplified 99.0%

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

9. Taylor expanded in maxCos around 0 99.0%

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

    1. mul-1-neg 99.0%

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

    2. sub-neg 99.0%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in ux around 0 99.0%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Final simplification 98.4%

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

Alternative 3: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.7999999499807018 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.7999999499807018e-6)
   (* (cos (* PI (* uy 2.0))) (sqrt (* ux (- 2.0 ux))))
   (* (sqrt (* ux (- 2.0 (* 2.0 maxCos)))) (cos (* 2.0 (* uy PI))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.7999999499807018e-6f) {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos)))) * cosf((2.0f * (uy * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.7999999499807018e-6))
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))) * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.7999999499807018e-6))
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))) * cos((single(2.0) * (uy * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.79999995e-6

   1. Initial program 61.4%

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

      1. associate-*l* 61.4%

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

      2. sub-neg 61.4%

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

      3. +-commutative 61.4%

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

      4. distribute-rgt-neg-in 61.4%

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

      5. fma-define 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in ux around 0 99.0%

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

   6. Taylor expanded in maxCos around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*r* 98.9%

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

      3. neg-mul-1 98.9%

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

      4. unsub-neg 98.9%

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

   8. Simplified 98.9%

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

   if 1.79999995e-6 < maxCos

   1. Initial program 44.5%

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

   3. Taylor expanded in ux around 0 84.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.7999999499807018 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if uy < 1.10000001e-4

   1. Initial program 59.6%

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

      1. associate-*l* 59.6%

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

      2. sub-neg 59.6%

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

      3. +-commutative 59.6%

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

      4. distribute-rgt-neg-in 59.6%

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

      5. fma-define 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in ux around inf 99.3%

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

   6. Taylor expanded in ux around 0 99.5%

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

      1. associate-+r+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. mul-1-neg 99.5%

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

      8. sub-neg 99.5%

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

      9. *-commutative 99.5%

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

      10. sub-neg 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in maxCos around 0 99.5%

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

       1. mul-1-neg 99.5%

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

       2. sub-neg 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in uy around 0 99.2%

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

   if 1.10000001e-4 < uy

   1. Initial program 58.4%

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

      1. associate-*l* 58.4%

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

      2. sub-neg 58.4%

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

      3. +-commutative 58.4%

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

      4. distribute-rgt-neg-in 58.4%

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

      5. fma-define 58.7%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in ux around 0 98.3%

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

   6. Taylor expanded in maxCos around 0 93.3%

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

      1. *-commutative 93.3%

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

      2. associate-*r* 93.3%

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

      3. neg-mul-1 93.3%

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

      4. unsub-neg 93.3%

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

   8. Simplified 93.3%

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

4. Final simplification 96.7%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in ux around 0 99.0%

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

6. Taylor expanded in maxCos around 0 97.8%

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

   1. neg-mul-1 97.8%

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

8. Simplified 97.8%

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

9. Final simplification 97.8%

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

Alternative 6: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.002090000081807375:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right) - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.002090000081807375)
   (sqrt
    (*
     ux
     (- (+ 2.0 (* ux (* (+ maxCos -1.0) (- 1.0 maxCos)))) (* 2.0 maxCos))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.002090000081807375f) {
		tmp = sqrtf((ux * ((2.0f + (ux * ((maxCos + -1.0f) * (1.0f - maxCos)))) - (2.0f * maxCos))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.002090000081807375))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.002090000081807375))
		tmp = sqrt((ux * ((single(2.0) + (ux * ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))) - (single(2.0) * maxCos))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00209000008

   1. Initial program 59.9%

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

      1. associate-*l* 59.9%

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

      2. sub-neg 59.9%

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

      3. +-commutative 59.9%

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

      4. distribute-rgt-neg-in 59.9%

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

      5. fma-define 60.2%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in ux around inf 99.2%

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

   6. Taylor expanded in ux around 0 99.3%

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

      1. associate-+r+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. distribute-lft-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. mul-1-neg 99.3%

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

      8. sub-neg 99.3%

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

      9. *-commutative 99.3%

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

      10. sub-neg 99.3%

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

      11. metadata-eval 99.3%

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

      12. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in maxCos around 0 99.4%

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

       1. mul-1-neg 99.4%

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

       2. sub-neg 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in uy around 0 96.3%

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

   if 0.00209000008 < uy

   1. Initial program 57.2%

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

      1. associate-*l* 57.2%

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

      2. sub-neg 57.2%

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

      3. +-commutative 57.2%

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

      4. distribute-rgt-neg-in 57.2%

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

      5. fma-define 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in ux around inf 98.2%

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

   6. Taylor expanded in maxCos around 0 92.8%

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

      1. sub-neg 92.8%

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

      2. associate-*r/ 92.8%

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

      3. metadata-eval 92.8%

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

      4. metadata-eval 92.8%

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

   8. Simplified 92.8%

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

   9. Taylor expanded in ux around 0 71.2%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.002090000081807375:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right) - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in ux around inf 98.9%

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

6. Taylor expanded in ux around 0 99.0%

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

   1. associate-+r+ 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. +-commutative 99.0%

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

   5. distribute-lft-in 99.0%

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

   6. metadata-eval 99.0%

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

   7. mul-1-neg 99.0%

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

   8. sub-neg 99.0%

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

   9. *-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. metadata-eval 99.0%

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

   12. +-commutative 99.0%

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

8. Simplified 99.0%

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

9. Taylor expanded in maxCos around 0 99.0%

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

    1. mul-1-neg 99.0%

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

    2. sub-neg 99.0%

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

11. Simplified 99.0%

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

12. Taylor expanded in uy around 0 79.9%

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

13. Final simplification 79.9%

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

Alternative 8: 79.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in uy around 0 50.5%

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

   1. mul-1-neg 50.5%

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

   2. unsub-neg 50.5%

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

   3. sub-neg 50.5%

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

   4. metadata-eval 50.5%

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

   5. distribute-lft-in 50.5%

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

   6. *-commutative 50.5%

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

   7. mul-1-neg 50.5%

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

   8. sub-neg 50.5%

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

   9. *-commutative 50.5%

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

   10. associate--l+ 50.5%

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

   11. unpow2 50.5%

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

   12. sub-neg 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in ux around 0 79.9%

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

   1. associate--l+ 80.0%

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

   2. associate-*r* 80.0%

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

   3. sub-neg 80.0%

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

   4. metadata-eval 80.0%

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

   5. +-commutative 80.0%

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

   6. neg-mul-1 80.0%

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

10. Simplified 80.0%

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

11. Taylor expanded in maxCos around 0 79.4%

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

    1. *-commutative 79.4%

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

13. Simplified 79.4%

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

14. Final simplification 79.4%

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

Alternative 9: 79.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in uy around 0 50.5%

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

   1. mul-1-neg 50.5%

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

   2. unsub-neg 50.5%

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

   3. sub-neg 50.5%

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

   4. metadata-eval 50.5%

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

   5. distribute-lft-in 50.5%

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

   6. *-commutative 50.5%

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

   7. mul-1-neg 50.5%

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

   8. sub-neg 50.5%

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

   9. *-commutative 50.5%

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

   10. associate--l+ 50.5%

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

   11. unpow2 50.5%

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

   12. sub-neg 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in ux around 0 79.9%

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

   1. associate--l+ 80.0%

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

   2. associate-*r* 80.0%

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

   3. sub-neg 80.0%

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

   4. metadata-eval 80.0%

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

   5. +-commutative 80.0%

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

   6. neg-mul-1 80.0%

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

10. Simplified 80.0%

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

11. Taylor expanded in maxCos around 0 79.0%

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

    1. neg-mul-1 79.0%

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

13. Simplified 79.0%

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

14. Final simplification 79.0%

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

Alternative 10: 75.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in ux around inf 98.9%

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

6. Taylor expanded in maxCos around 0 92.9%

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

   1. sub-neg 92.9%

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

   2. associate-*r/ 92.9%

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

   3. metadata-eval 92.9%

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

   4. metadata-eval 92.9%

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

8. Simplified 92.9%

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

9. Taylor expanded in uy around 0 75.5%

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

    1. sub-neg 75.5%

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

    2. associate-*r/ 75.5%

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

    3. metadata-eval 75.5%

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

    4. metadata-eval 75.5%

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

11. Simplified 75.5%

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

12. Final simplification 75.5%

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

Alternative 11: 65.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in uy around 0 50.5%

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

   1. mul-1-neg 50.5%

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

   2. unsub-neg 50.5%

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

   3. sub-neg 50.5%

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

   4. metadata-eval 50.5%

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

   5. distribute-lft-in 50.5%

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

   6. *-commutative 50.5%

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

   7. mul-1-neg 50.5%

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

   8. sub-neg 50.5%

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

   9. *-commutative 50.5%

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

   10. associate--l+ 50.5%

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

   11. unpow2 50.5%

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

   12. sub-neg 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in ux around 0 63.4%

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

Alternative 12: 30.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in uy around 0 50.5%

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

   1. mul-1-neg 50.5%

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

   2. unsub-neg 50.5%

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

   3. sub-neg 50.5%

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

   4. metadata-eval 50.5%

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

   5. distribute-lft-in 50.5%

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

   6. *-commutative 50.5%

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

   7. mul-1-neg 50.5%

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

   8. sub-neg 50.5%

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

   9. *-commutative 50.5%

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

   10. associate--l+ 50.5%

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

   11. unpow2 50.5%

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

   12. sub-neg 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in maxCos around inf 25.2%

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

9. Taylor expanded in maxCos around 0 30.3%

   \[\leadsto \sqrt{\color{blue}{ux}} \]
10. Add Preprocessing

Alternative 13: 18.6% accurate, 31.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in uy around 0 50.5%

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

   1. mul-1-neg 50.5%

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

   2. unsub-neg 50.5%

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

   3. sub-neg 50.5%

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

   4. metadata-eval 50.5%

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

   5. distribute-lft-in 50.5%

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

   6. *-commutative 50.5%

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

   7. mul-1-neg 50.5%

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

   8. sub-neg 50.5%

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

   9. *-commutative 50.5%

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

   10. associate--l+ 50.5%

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

   11. unpow2 50.5%

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

   12. sub-neg 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in maxCos around inf -0.0%

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

10. Applied egg-rr 18.8%

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

    1. sub-neg 18.8%

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

    2. log1p-undefine 18.8%

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

    3. rem-exp-log 18.8%

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

    4. metadata-eval 18.8%

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

12. Simplified 18.8%

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

13. Final simplification 18.8%

    \[\leadsto 1 + \left(1 + ux \cdot maxCos\right) \]
14. Add Preprocessing

Alternative 14: 12.4% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \frac{1}{ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* maxCos (/ 1.0 ux)))

C

float code(float ux, float uy, float maxCos) {
	return maxCos * (1.0f / ux);
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (1.0e0 / ux)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(maxCos * Float32(Float32(1.0) / ux))
end

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in uy around 0 50.5%

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

   1. mul-1-neg 50.5%

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

   2. unsub-neg 50.5%

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

   3. sub-neg 50.5%

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

   4. metadata-eval 50.5%

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

   5. distribute-lft-in 50.5%

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

   6. *-commutative 50.5%

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

   7. mul-1-neg 50.5%

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

   8. sub-neg 50.5%

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

   9. *-commutative 50.5%

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

   10. associate--l+ 50.5%

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

   11. unpow2 50.5%

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

   12. sub-neg 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in maxCos around inf -0.0%

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

10. Applied egg-rr 12.1%

    \[\leadsto maxCos \cdot \color{blue}{{ux}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 12.1%

       \[\leadsto maxCos \cdot \color{blue}{\frac{1}{ux}} \]

12. Simplified 12.1%

    \[\leadsto maxCos \cdot \color{blue}{\frac{1}{ux}} \]
13. Add Preprocessing

Alternative 15: 9.6% accurate, 74.3× speedup

Math

\[\begin{array}{l} \\ ux \cdot maxCos \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return ux * 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 = ux * maxcos
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.1%

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

   1. associate-*l* 59.1%

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

   2. sub-neg 59.1%

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

   3. +-commutative 59.1%

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

   4. distribute-rgt-neg-in 59.1%

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

   5. fma-define 59.5%

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

3. Simplified 59.5%

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

5. Taylor expanded in uy around 0 50.5%

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

   1. mul-1-neg 50.5%

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

   2. unsub-neg 50.5%

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

   3. sub-neg 50.5%

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

   4. metadata-eval 50.5%

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

   5. distribute-lft-in 50.5%

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

   6. *-commutative 50.5%

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

   7. mul-1-neg 50.5%

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

   8. sub-neg 50.5%

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

   9. *-commutative 50.5%

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

   10. associate--l+ 50.5%

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

   11. unpow2 50.5%

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

   12. sub-neg 50.5%

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

7. Simplified 50.5%

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

8. Taylor expanded in maxCos around inf -0.0%

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

10. Applied egg-rr 9.7%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{maxCos \cdot ux}\right)}^{3}} \]
11. Step-by-step derivation

    1. rem-cube-cbrt 9.7%

       \[\leadsto \color{blue}{maxCos \cdot ux} \]

12. Simplified 9.7%

    \[\leadsto \color{blue}{maxCos \cdot ux} \]

13. Final simplification 9.7%

    \[\leadsto ux \cdot maxCos \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024172 
(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)))))))