UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 25.0s

Alternatives: 10

Speedup: 0.9×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t_0 \cdot t_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t_2 \cdot t_1\right) \cdot xi + \left(\sin t_2 \cdot t_1\right) \cdot yi\right) + t_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t_0 \cdot t_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t_2 \cdot t_1\right) \cdot xi + \left(\sin t_2 \cdot t_1\right) \cdot yi\right) + t_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right), 1\right)}, \mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right), xi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right), \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)} \]
4. Step-by-step derivation

   1. add-log-exp 98.9%

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

   2. *-commutative 98.9%

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

   3. exp-prod 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot maxCos\\ t_1 := uy \cdot \left(2 \cdot \pi\right)\\ t_2 := \sqrt{1 + \left(ux \cdot t_0\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\\ \mathsf{fma}\left(\cos t_1 \cdot t_2, xi, \sin t_1 \cdot \left(yi \cdot t_2\right)\right) + t_0 \cdot \left(ux \cdot zi\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) maxCos))
        (t_1 (* uy (* 2.0 PI)))
        (t_2 (sqrt (+ 1.0 (* (* ux t_0) (* ux (* maxCos (+ ux -1.0))))))))
   (+ (fma (* (cos t_1) t_2) xi (* (sin t_1) (* yi t_2))) (* t_0 (* ux zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * maxCos;
	float t_1 = uy * (2.0f * ((float) M_PI));
	float t_2 = sqrtf((1.0f + ((ux * t_0) * (ux * (maxCos * (ux + -1.0f))))));
	return fmaf((cosf(t_1) * t_2), xi, (sinf(t_1) * (yi * t_2))) + (t_0 * (ux * zi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * maxCos)
	t_1 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	t_2 = sqrt(Float32(Float32(1.0) + Float32(Float32(ux * t_0) * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0)))))))
	return Float32(fma(Float32(cos(t_1) * t_2), xi, Float32(sin(t_1) * Float32(yi * t_2))) + Float32(t_0 * Float32(ux * zi)))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot maxCos\\ t_0 \cdot \left(ux \cdot zi\right) + \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 + \left(ux \cdot t_0\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}, xi, yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) maxCos)))
   (+
    (* t_0 (* ux zi))
    (fma
     (*
      (cos (* uy (* 2.0 PI)))
      (sqrt (+ 1.0 (* (* ux t_0) (* ux (* maxCos (+ ux -1.0)))))))
     xi
     (* yi (sin (* PI (* 2.0 uy))))))))

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in ux around 0 98.9%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)}, xi, \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) + \left(\left(1 - ux\right) \cdot maxCos\right) \cdot \left(ux \cdot zi\right) \]
5. Step-by-step derivation

   1. associate-*r* 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot \left(maxCos \cdot \left(1 + ux \cdot -2\right)\right)\right)\right)} \cdot \left(xi \cdot \cos t_0 + yi \cdot \sin t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (fma
    ux
    (* (- 1.0 ux) (* maxCos zi))
    (*
     (sqrt (- 1.0 (* ux (* ux (* maxCos (* maxCos (+ 1.0 (* ux -2.0))))))))
     (+ (* xi (cos t_0)) (* yi (sin t_0)))))))

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in ux around 0 98.6%

   \[\leadsto \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot \left(maxCos \cdot \color{blue}{\left(1 + -2 \cdot ux\right)}\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \left(xi \cdot \cos t_0 + yi \cdot \sin t_0\right) \cdot \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (fma
    ux
    (* (- 1.0 ux) (* maxCos zi))
    (*
     (+ (* xi (cos t_0)) (* yi (sin t_0)))
     (sqrt (- 1.0 (* ux (* ux (* maxCos maxCos)))))))))

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in ux around 0 98.5%

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

5. Final simplification 98.5%

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

Alternative 6: 88.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot maxCos\\ t_0 \cdot \left(ux \cdot zi\right) + \mathsf{fma}\left(\sqrt{1 + \left(ux \cdot t_0\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}, xi, yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) maxCos)))
   (+
    (* t_0 (* ux zi))
    (fma
     (sqrt (+ 1.0 (* (* ux t_0) (* ux (* maxCos (+ ux -1.0))))))
     xi
     (* yi (sin (* PI (* 2.0 uy))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * maxCos;
	return (t_0 * (ux * zi)) + fmaf(sqrtf((1.0f + ((ux * t_0) * (ux * (maxCos * (ux + -1.0f)))))), xi, (yi * sinf((((float) M_PI) * (2.0f * uy)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * maxCos)
	return Float32(Float32(t_0 * Float32(ux * zi)) + fma(sqrt(Float32(Float32(1.0) + Float32(Float32(ux * t_0) * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0))))))), xi, Float32(yi * sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in ux around 0 98.9%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)}, xi, \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) + \left(\left(1 - ux\right) \cdot maxCos\right) \cdot \left(ux \cdot zi\right) \]
5. Step-by-step derivation

   1. associate-*r* 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around 0 89.3%

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

8. Final simplification 89.3%

   \[\leadsto \left(\left(1 - ux\right) \cdot maxCos\right) \cdot \left(ux \cdot zi\right) + \mathsf{fma}\left(\sqrt{1 + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}, xi, yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \]

Alternative 7: 81.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  ux
  (- (* maxCos zi) (* ux (* maxCos zi)))
  (*
   (sqrt (- 1.0 (* ux (* ux (* maxCos maxCos)))))
   (+ xi (* 2.0 (* PI (* uy yi)))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(ux, Float32(Float32(maxCos * zi) - Float32(ux * Float32(maxCos * zi))), Float32(sqrt(Float32(Float32(1.0) - Float32(ux * Float32(ux * Float32(maxCos * maxCos))))) * Float32(xi + Float32(Float32(2.0) * Float32(Float32(pi) * Float32(uy * yi))))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in uy around 0 87.5%

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

   1. associate-*r* 87.5%

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

6. Simplified 87.5%

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

7. Taylor expanded in ux around 0 87.2%

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

8. Taylor expanded in uy around 0 79.3%

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

   1. *-commutative 79.3%

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

   2. sub-neg 79.3%

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

   3. distribute-lft-in 79.4%

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

   4. *-commutative 79.4%

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

   5. *-un-lft-identity 79.4%

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

10. Applied egg-rr 79.4%

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

11. Final simplification 79.4%

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

Alternative 8: 81.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  ux
  (* (- 1.0 ux) (* maxCos zi))
  (*
   (+ xi (* 2.0 (* PI (* uy yi))))
   (sqrt (- 1.0 (* (* maxCos maxCos) (* ux ux)))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(ux, Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * zi)), Float32(Float32(xi + Float32(Float32(2.0) * Float32(Float32(pi) * Float32(uy * yi)))) * sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(ux * ux))))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in uy around 0 87.5%

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

   1. associate-*r* 87.5%

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

6. Simplified 87.5%

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

7. Taylor expanded in ux around 0 87.2%

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

8. Taylor expanded in uy around 0 79.3%

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

9. Taylor expanded in ux around 0 79.3%

   \[\leadsto \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - \color{blue}{{maxCos}^{2} \cdot {ux}^{2}}} \cdot \left(1 \cdot xi + 2 \cdot \left(\left(uy \cdot yi\right) \cdot \pi\right)\right)\right) \]
10. Step-by-step derivation

    1. unpow2 79.3%

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

    2. unpow2 79.3%

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

11. Simplified 79.3%

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

12. Final simplification 79.3%

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

Alternative 9: 81.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  ux
  (* maxCos (- zi (* ux zi)))
  (*
   (sqrt (- 1.0 (* ux (* ux (* maxCos maxCos)))))
   (+ xi (* 2.0 (* PI (* uy yi)))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(ux, Float32(maxCos * Float32(zi - Float32(ux * zi))), Float32(sqrt(Float32(Float32(1.0) - Float32(ux * Float32(ux * Float32(maxCos * maxCos))))) * Float32(xi + Float32(Float32(2.0) * Float32(Float32(pi) * Float32(uy * yi))))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in uy around 0 87.5%

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

   1. associate-*r* 87.5%

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

6. Simplified 87.5%

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

7. Taylor expanded in ux around 0 87.2%

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

8. Taylor expanded in uy around 0 79.3%

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

9. Taylor expanded in ux around 0 79.4%

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

    1. +-commutative 79.4%

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

    2. mul-1-neg 79.4%

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

    3. unsub-neg 79.4%

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

    4. distribute-lft-out-- 79.4%

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

    5. *-commutative 79.4%

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

11. Simplified 79.4%

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

12. Final simplification 79.4%

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

Alternative 10: 79.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  ux
  (* maxCos zi)
  (*
   (sqrt (- 1.0 (* ux (* ux (* maxCos maxCos)))))
   (+ xi (* 2.0 (* PI (* uy yi)))))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(ux, Float32(maxCos * zi), Float32(sqrt(Float32(Float32(1.0) - Float32(ux * Float32(ux * Float32(maxCos * maxCos))))) * Float32(xi + Float32(Float32(2.0) * Float32(Float32(pi) * Float32(uy * yi))))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in uy around 0 87.5%

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

   1. associate-*r* 87.5%

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

6. Simplified 87.5%

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

7. Taylor expanded in ux around 0 87.2%

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

8. Taylor expanded in uy around 0 79.3%

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

9. Taylor expanded in ux around 0 76.2%

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

10. Final simplification 76.2%

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

Reproduce

herbie shell --seed 2023200 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (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) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))