UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 23.3s

Alternatives: 22

Speedup: 1.0×

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\\ t_1 := \sqrt{1 + t\_0 \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(t\_1 \cdot \sin t\_2\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Final simplification 98.8%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(\left(1 - ux\right) \cdot maxCos\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathsf{fma}\left(1 - ux, zi \cdot \left(ux \cdot maxCos\right), \mathsf{fma}\left(\cos t\_0, xi \cdot t\_1, t\_1 \cdot \left(yi \cdot \sin t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, t\_1 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), t\_1 \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, t\_1, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI)))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.012000000104308128)
     (fma
      uy
      (fma
       uy
       (*
        t_1
        (fma
         -2.0
         (* xi (* PI PI))
         (* -1.3333333333333333 (* uy (* yi (* PI (* PI PI)))))))
       (* t_1 (* 2.0 (* PI yi))))
      (fma xi t_1 (* maxCos (* (- 1.0 ux) (* ux zi)))))
     (fma maxCos (* ux zi) (fma yi (sin t_0) (* xi (cos t_0)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.012000000104308128f) {
		tmp = fmaf(uy, fmaf(uy, (t_1 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))))), (t_1 * (2.0f * (((float) M_PI) * yi)))), fmaf(xi, t_1, (maxCos * ((1.0f - ux) * (ux * zi)))));
	} else {
		tmp = fmaf(maxCos, (ux * zi), fmaf(yi, sinf(t_0), (xi * cosf(t_0))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.012000000104308128))
		tmp = fma(uy, fma(uy, Float32(t_1 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))))), Float32(t_1 * Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), fma(xi, t_1, Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)))));
	else
		tmp = fma(maxCos, Float32(ux * zi), fma(yi, sin(t_0), Float32(xi * cos(t_0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0120000001

   1. Initial program 99.1%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(uy \cdot \left(2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + uy \cdot \left(-2 \cdot \left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \frac{-4}{3} \cdot \left(\left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)}\right), \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)}, maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)} \]

   if 0.0120000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.5%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f32 N/A

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

      5. add-cube-cbrt N/A

         \[\leadsto \left(\left(\cos \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. lower-*.f32 N/A

         \[\leadsto \left(\left(\cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. pow2 N/A

         \[\leadsto \left(\left(\cos \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. lift-PI.f32 N/A

         \[\leadsto \left(\left(\cos \left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      10. pow1/3 N/A

          \[\leadsto \left(\left(\cos \left({\color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{3}}\right)}}^{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      11. pow-pow N/A

          \[\leadsto \left(\left(\cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{3} \cdot 2\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      12. lower-pow.f32 N/A

          \[\leadsto \left(\left(\cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{3} \cdot 2\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      14. lower-*.f32 N/A

          \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      15. lift-PI.f32 N/A

          \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      16. lower-cbrt.f32 97.3

          \[\leadsto \left(\left(\cos \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied rewrites 97.3%

      \[\leadsto \left(\left(\cos \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 95.8

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

   7. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(xi, \cos t\_0, \mathsf{fma}\left(yi, \sin t\_0, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-cos.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-sin.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 N/A

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

5. Applied rewrites 98.6%

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

6. Final simplification 98.6%

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

Alternative 5: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, t\_1 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), t\_1 \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, t\_1, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI)))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.012000000104308128)
     (fma
      uy
      (fma
       uy
       (*
        t_1
        (fma
         -2.0
         (* xi (* PI PI))
         (* -1.3333333333333333 (* uy (* yi (* PI (* PI PI)))))))
       (* t_1 (* 2.0 (* PI yi))))
      (fma xi t_1 (* maxCos (* (- 1.0 ux) (* ux zi)))))
     (fma yi (sin t_0) (* xi (cos t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.012000000104308128f) {
		tmp = fmaf(uy, fmaf(uy, (t_1 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))))), (t_1 * (2.0f * (((float) M_PI) * yi)))), fmaf(xi, t_1, (maxCos * ((1.0f - ux) * (ux * zi)))));
	} else {
		tmp = fmaf(yi, sinf(t_0), (xi * cosf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.012000000104308128))
		tmp = fma(uy, fma(uy, Float32(t_1 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))))), Float32(t_1 * Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), fma(xi, t_1, Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)))));
	else
		tmp = fma(yi, sin(t_0), Float32(xi * cos(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0120000001

   1. Initial program 99.1%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(uy \cdot \left(2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + uy \cdot \left(-2 \cdot \left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \frac{-4}{3} \cdot \left(\left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)}\right), \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)}, maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)} \]

   if 0.0120000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.5%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f32 N/A

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

      5. add-cube-cbrt N/A

         \[\leadsto \left(\left(\cos \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. lower-*.f32 N/A

         \[\leadsto \left(\left(\cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. pow2 N/A

         \[\leadsto \left(\left(\cos \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. lift-PI.f32 N/A

         \[\leadsto \left(\left(\cos \left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      10. pow1/3 N/A

          \[\leadsto \left(\left(\cos \left({\color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{3}}\right)}}^{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      11. pow-pow N/A

          \[\leadsto \left(\left(\cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{3} \cdot 2\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      12. lower-pow.f32 N/A

          \[\leadsto \left(\left(\cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{3} \cdot 2\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      13. metadata-eval N/A

          \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      14. lower-*.f32 N/A

          \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      15. lift-PI.f32 N/A

          \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      16. lower-cbrt.f32 97.3

          \[\leadsto \left(\left(\cos \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied rewrites 97.3%

      \[\leadsto \left(\left(\cos \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      3. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi, \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi, \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi, \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      8. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), xi \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), xi \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), xi \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      11. lower-PI.f32 94.1

          \[\leadsto \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right)\right) \]

   7. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, t\_1 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), t\_1 \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, t\_1, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI)))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.012000000104308128)
     (fma
      uy
      (fma
       uy
       (*
        t_1
        (fma
         -2.0
         (* xi (* PI PI))
         (* -1.3333333333333333 (* uy (* yi (* PI (* PI PI)))))))
       (* t_1 (* 2.0 (* PI yi))))
      (fma xi t_1 (* maxCos (* (- 1.0 ux) (* ux zi)))))
     (fma 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 = 2.0f * (uy * ((float) M_PI));
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.012000000104308128f) {
		tmp = fmaf(uy, fmaf(uy, (t_1 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))))), (t_1 * (2.0f * (((float) M_PI) * yi)))), fmaf(xi, t_1, (maxCos * ((1.0f - ux) * (ux * zi)))));
	} else {
		tmp = fmaf(xi, cosf(t_0), (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.012000000104308128))
		tmp = fma(uy, fma(uy, Float32(t_1 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))))), Float32(t_1 * Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), fma(xi, t_1, Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)))));
	else
		tmp = fma(xi, cos(t_0), Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0120000001

   1. Initial program 99.1%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(uy \cdot \left(2 \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + uy \cdot \left(-2 \cdot \left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) + \frac{-4}{3} \cdot \left(\left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)}\right), \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)}, maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)} \]

   if 0.0120000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.5%

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

   3. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      10. lower-PI.f32 94.1

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right)\right) \]

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - ux\right)\\ zi \cdot \mathsf{fma}\left(maxCos, t\_0, \sqrt{\mathsf{fma}\left(t\_0 \cdot t\_0, maxCos \cdot \left(-maxCos\right), 1\right)} \cdot \mathsf{fma}\left(xi, \frac{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}, \frac{\left(uy \cdot 2\right) \cdot \left(\pi \cdot yi\right)}{zi}\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-PI.f32 N/A

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

   5. add-cube-cbrt N/A

      \[\leadsto \left(\left(\cos \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   6. associate-*l* N/A

      \[\leadsto \left(\left(\cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. pow2 N/A

      \[\leadsto \left(\left(\cos \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   9. lift-PI.f32 N/A

      \[\leadsto \left(\left(\cos \left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   10. pow1/3 N/A

       \[\leadsto \left(\left(\cos \left({\color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{3}}\right)}}^{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   11. pow-pow N/A

       \[\leadsto \left(\left(\cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{3} \cdot 2\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   12. lower-pow.f32 N/A

       \[\leadsto \left(\left(\cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{3} \cdot 2\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   14. lower-*.f32 N/A

       \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(uy \cdot 2\right)\right)}\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   15. lift-PI.f32 N/A

       \[\leadsto \left(\left(\cos \left({\mathsf{PI}\left(\right)}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   16. lower-cbrt.f32 98.7

       \[\leadsto \left(\left(\cos \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied rewrites 98.7%

   \[\leadsto \left(\left(\cos \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(uy \cdot 2\right)\right)\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Taylor expanded in zi around inf

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in uy around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 90.4

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

10. Applied rewrites 90.4%

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

11. Final simplification 90.4%

    \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \sqrt{\mathsf{fma}\left(\left(ux \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(1 - ux\right)\right), maxCos \cdot \left(-maxCos\right), 1\right)} \cdot \mathsf{fma}\left(xi, \frac{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}, \frac{\left(uy \cdot 2\right) \cdot \left(\pi \cdot yi\right)}{zi}\right)\right) \]
12. Add Preprocessing

Alternative 8: 90.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - ux\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.15000000596046448:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)}{xi}, -2 \cdot \left(\pi \cdot \pi\right)\right), \frac{2 \cdot \left(\pi \cdot yi\right)}{xi}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \mathsf{fma}\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}, \sqrt{\mathsf{fma}\left(t\_0 \cdot t\_0, maxCos \cdot \left(-maxCos\right), 1\right)}, maxCos \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 1.0 ux))))
   (if (<= (* uy 2.0) 0.15000000596046448)
     (+
      (* (* ux (* (- 1.0 ux) maxCos)) zi)
      (*
       xi
       (*
        (sqrt
         (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
        (fma
         uy
         (fma
          uy
          (fma
           -1.3333333333333333
           (/ (* (* PI (* PI PI)) (* uy yi)) xi)
           (* -2.0 (* PI PI)))
          (/ (* 2.0 (* PI yi)) xi))
         1.0))))
     (*
      zi
      (fma
       (/ (* xi (cos (* 2.0 (* uy PI)))) zi)
       (sqrt (fma (* t_0 t_0) (* maxCos (- maxCos)) 1.0))
       (* maxCos t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - ux);
	float tmp;
	if ((uy * 2.0f) <= 0.15000000596046448f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + (xi * (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf(uy, fmaf(uy, fmaf(-1.3333333333333333f, (((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)) / xi), (-2.0f * (((float) M_PI) * ((float) M_PI)))), ((2.0f * (((float) M_PI) * yi)) / xi)), 1.0f)));
	} else {
		tmp = zi * fmaf(((xi * cosf((2.0f * (uy * ((float) M_PI))))) / zi), sqrtf(fmaf((t_0 * t_0), (maxCos * -maxCos), 1.0f)), (maxCos * t_0));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.15000000596046448))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(xi * Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(uy, fma(uy, fma(Float32(-1.3333333333333333), Float32(Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi)) / xi), Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(Float32(2.0) * Float32(Float32(pi) * yi)) / xi)), Float32(1.0)))));
	else
		tmp = Float32(zi * fma(Float32(Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) / zi), sqrt(fma(Float32(t_0 * t_0), Float32(maxCos * Float32(-maxCos)), Float32(1.0))), Float32(maxCos * t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.150000006

   1. Initial program 99.1%

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

   3. Taylor expanded in xi around inf

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

      1. lower-*.f32 N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in uy around 0

      \[\leadsto xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \color{blue}{\left(1 + uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \color{blue}{\left(uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right) + 1\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. lower-fma.f32 N/A

         \[\leadsto xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \color{blue}{\mathsf{fma}\left(uy, 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right), 1\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. Applied rewrites 96.9%

      \[\leadsto xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \color{blue}{\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \frac{\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{xi}, -2 \cdot \left(\pi \cdot \pi\right)\right), \frac{2 \cdot \left(yi \cdot \pi\right)}{xi}\right), 1\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.150000006 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 96.3%

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

   3. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in zi around inf

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 59.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.15000000596046448:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)}{xi}, -2 \cdot \left(\pi \cdot \pi\right)\right), \frac{2 \cdot \left(\pi \cdot yi\right)}{xi}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \mathsf{fma}\left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}, \sqrt{\mathsf{fma}\left(\left(ux \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(1 - ux\right)\right), maxCos \cdot \left(-maxCos\right), 1\right)}, maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.15000000596046448:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)}{xi}, -2 \cdot \left(\pi \cdot \pi\right)\right), \frac{2 \cdot \left(\pi \cdot yi\right)}{xi}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.15000000596046448)
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) zi)
    (*
     xi
     (*
      (sqrt
       (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
      (fma
       uy
       (fma
        uy
        (fma
         -1.3333333333333333
         (/ (* (* PI (* PI PI)) (* uy yi)) xi)
         (* -2.0 (* PI PI)))
        (/ (* 2.0 (* PI yi)) xi))
       1.0))))
   (fma xi (cos (* 2.0 (* uy PI))) (* maxCos (* (- 1.0 ux) (* ux zi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.15000000596046448f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + (xi * (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf(uy, fmaf(uy, fmaf(-1.3333333333333333f, (((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)) / xi), (-2.0f * (((float) M_PI) * ((float) M_PI)))), ((2.0f * (((float) M_PI) * yi)) / xi)), 1.0f)));
	} else {
		tmp = fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), (maxCos * ((1.0f - ux) * (ux * zi))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.15000000596046448))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(xi * Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(uy, fma(uy, fma(Float32(-1.3333333333333333), Float32(Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi)) / xi), Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(Float32(2.0) * Float32(Float32(pi) * yi)) / xi)), Float32(1.0)))));
	else
		tmp = fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.150000006

   1. Initial program 99.1%

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

   3. Taylor expanded in xi around inf

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

      1. lower-*.f32 N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in uy around 0

      \[\leadsto xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \color{blue}{\left(1 + uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right)\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \color{blue}{\left(uy \cdot \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right)\right) + 1\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. lower-fma.f32 N/A

         \[\leadsto xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \color{blue}{\mathsf{fma}\left(uy, 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi} + uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-4}{3} \cdot \frac{uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)}{xi}\right), 1\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. Applied rewrites 96.9%

      \[\leadsto xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \color{blue}{\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \frac{\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)}{xi}, -2 \cdot \left(\pi \cdot \pi\right)\right), \frac{2 \cdot \left(yi \cdot \pi\right)}{xi}\right), 1\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.150000006 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 96.3%

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

   3. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 59.5

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

   8. Applied rewrites 59.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.15000000596046448:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)}{xi}, -2 \cdot \left(\pi \cdot \pi\right)\right), \frac{2 \cdot \left(\pi \cdot yi\right)}{xi}\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.1120000034570694:\\ \;\;\;\;\mathsf{fma}\left(xi, t\_1, \mathsf{fma}\left(uy, t\_1 \cdot \mathsf{fma}\left(-2, \left(\pi \cdot \pi\right) \cdot \left(uy \cdot xi\right), 2 \cdot \left(\pi \cdot yi\right)\right), t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* maxCos (* (- 1.0 ux) (* ux zi))))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.1120000034570694)
     (fma
      xi
      t_1
      (fma
       uy
       (* t_1 (fma -2.0 (* (* PI PI) (* uy xi)) (* 2.0 (* PI yi))))
       t_0))
     (fma xi (cos (* 2.0 (* uy PI))) t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * ((1.0f - ux) * (ux * zi));
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.1120000034570694f) {
		tmp = fmaf(xi, t_1, fmaf(uy, (t_1 * fmaf(-2.0f, ((((float) M_PI) * ((float) M_PI)) * (uy * xi)), (2.0f * (((float) M_PI) * yi)))), t_0));
	} else {
		tmp = fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), t_0);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)))
	t_1 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.1120000034570694))
		tmp = fma(xi, t_1, fma(uy, Float32(t_1 * fma(Float32(-2.0), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(uy * xi)), Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), t_0));
	else
		tmp = fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.112000003

   1. Initial program 99.1%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 95.0%

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

   if 0.112000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 96.4%

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

   3. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 59.1

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

   8. Applied rewrites 59.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.1120000034570694:\\ \;\;\;\;\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, \left(\pi \cdot \pi\right) \cdot \left(uy \cdot xi\right), 2 \cdot \left(\pi \cdot yi\right)\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.1120000034570694:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(uy, \mathsf{fma}\left(uy \cdot -2, xi \cdot \left(\pi \cdot \pi\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* maxCos (* (- 1.0 ux) (* ux zi)))))
   (if (<= (* uy 2.0) 0.1120000034570694)
     (fma
      (sqrt
       (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
      (fma uy (fma (* uy -2.0) (* xi (* PI PI)) (* 2.0 (* PI yi))) xi)
      t_0)
     (fma xi (cos (* 2.0 (* uy PI))) t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = maxCos * ((1.0f - ux) * (ux * zi));
	float tmp;
	if ((uy * 2.0f) <= 0.1120000034570694f) {
		tmp = fmaf(sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)), fmaf(uy, fmaf((uy * -2.0f), (xi * (((float) M_PI) * ((float) M_PI))), (2.0f * (((float) M_PI) * yi))), xi), t_0);
	} else {
		tmp = fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), t_0);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.1120000034570694))
		tmp = fma(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))), fma(uy, fma(Float32(uy * Float32(-2.0)), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(Float32(pi) * yi))), xi), t_0);
	else
		tmp = fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.112000003

   1. Initial program 99.1%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in zi around 0

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

   8. Applied rewrites 95.0%

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

   if 0.112000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 96.4%

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

   3. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 59.1

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

   8. Applied rewrites 59.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.1120000034570694:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(uy, \mathsf{fma}\left(uy \cdot -2, xi \cdot \left(\pi \cdot \pi\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.1120000034570694:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(uy, \mathsf{fma}\left(uy \cdot -2, xi \cdot \left(\pi \cdot \pi\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(ux \cdot zi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.1120000034570694)
   (fma
    (sqrt (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
    (fma uy (fma (* uy -2.0) (* xi (* PI PI)) (* 2.0 (* PI yi))) xi)
    (* maxCos (* (- 1.0 ux) (* ux zi))))
   (fma xi (cos (* 2.0 (* uy PI))) (* maxCos (* ux zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.1120000034570694f) {
		tmp = fmaf(sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)), fmaf(uy, fmaf((uy * -2.0f), (xi * (((float) M_PI) * ((float) M_PI))), (2.0f * (((float) M_PI) * yi))), xi), (maxCos * ((1.0f - ux) * (ux * zi))));
	} else {
		tmp = fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), (maxCos * (ux * zi)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.1120000034570694))
		tmp = fma(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))), fma(uy, fma(Float32(uy * Float32(-2.0)), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(Float32(pi) * yi))), xi), Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi))));
	else
		tmp = fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(maxCos * Float32(ux * zi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.112000003

   1. Initial program 99.1%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in zi around 0

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

   8. Applied rewrites 95.0%

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

   if 0.112000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 96.4%

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

   3. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(ux \cdot zi\right)\right)} \]

      3. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, maxCos \cdot \left(ux \cdot zi\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, maxCos \cdot \left(ux \cdot zi\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), maxCos \cdot \left(ux \cdot zi\right)\right) \]

      6. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), maxCos \cdot \left(ux \cdot zi\right)\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}\right) \]

      8. lower-*.f32 57.0

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

   8. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(ux \cdot zi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.1120000034570694:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(uy, \mathsf{fma}\left(uy \cdot -2, xi \cdot \left(\pi \cdot \pi\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(ux \cdot zi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.15000000596046448:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(uy, \mathsf{fma}\left(uy \cdot -2, xi \cdot \left(\pi \cdot \pi\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.15000000596046448)
   (fma
    (sqrt (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
    (fma uy (fma (* uy -2.0) (* xi (* PI PI)) (* 2.0 (* PI yi))) xi)
    (* maxCos (* (- 1.0 ux) (* ux zi))))
   (* xi (cos (* 2.0 (* uy PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.15000000596046448f) {
		tmp = fmaf(sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)), fmaf(uy, fmaf((uy * -2.0f), (xi * (((float) M_PI) * ((float) M_PI))), (2.0f * (((float) M_PI) * yi))), xi), (maxCos * ((1.0f - ux) * (ux * zi))));
	} else {
		tmp = xi * cosf((2.0f * (uy * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.15000000596046448))
		tmp = fma(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))), fma(uy, fma(Float32(uy * Float32(-2.0)), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(Float32(pi) * yi))), xi), Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi))));
	else
		tmp = Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.150000006

   1. Initial program 99.1%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in zi around 0

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

   8. Applied rewrites 94.6%

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

   if 0.150000006 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 96.3%

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

   3. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. lower-cos.f32 N/A

         \[\leadsto xi \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto xi \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto xi \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      5. lower-PI.f32 56.6

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

   8. Applied rewrites 56.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.15000000596046448:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(uy, \mathsf{fma}\left(uy \cdot -2, xi \cdot \left(\pi \cdot \pi\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 86.8%

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

6. Taylor expanded in zi around 0

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

8. Applied rewrites 86.8%

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

9. Final simplification 86.8%

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

Alternative 15: 85.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 86.8%

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

6. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. lower-*.f32 N/A

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

8. Applied rewrites 86.7%

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

9. Final simplification 86.7%

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

Alternative 16: 82.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 86.8%

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

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{xi + \left(maxCos \cdot \left(ux \cdot zi\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

8. Applied rewrites 83.1%

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

9. Final simplification 83.1%

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

Alternative 17: 81.2% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 81.9%

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

9. Final simplification 81.9%

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

Alternative 18: 81.2% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 81.9

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

8. Applied rewrites 81.9%

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

9. Final simplification 81.9%

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

Alternative 19: 78.5% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.1%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 82.0

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in ux around 0

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

   1. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

   2. lower-+.f32 N/A

      \[\leadsto \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)} + maxCos \cdot \left(ux \cdot zi\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\color{blue}{\left(2 \cdot uy\right) \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)} + xi\right) + maxCos \cdot \left(ux \cdot zi\right) \]

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 78.6

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

10. Applied rewrites 78.6%

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

11. Final simplification 78.6%

    \[\leadsto maxCos \cdot \left(ux \cdot zi\right) + \mathsf{fma}\left(uy \cdot 2, \pi \cdot yi, xi\right) \]
12. Add Preprocessing

Alternative 20: 73.7% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(uy \cdot 2, \pi \cdot yi, xi\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot uy\right) \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)} + xi \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot uy, yi \cdot \mathsf{PI}\left(\right), xi\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{2 \cdot uy}, yi \cdot \mathsf{PI}\left(\right), xi\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(2 \cdot uy, \color{blue}{yi \cdot \mathsf{PI}\left(\right)}, xi\right) \]

   6. lower-PI.f32 73.8

      \[\leadsto \mathsf{fma}\left(2 \cdot uy, yi \cdot \color{blue}{\pi}, xi\right) \]

8. Applied rewrites 73.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot uy, yi \cdot \pi, xi\right)} \]

9. Final simplification 73.8%

   \[\leadsto \mathsf{fma}\left(uy \cdot 2, \pi \cdot yi, xi\right) \]
10. Add Preprocessing

Alternative 21: 11.9% accurate, 32.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* ux (* maxCos zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ux * (maxCos * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * (maxcos * zi)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in zi around inf

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

   1. lower-*.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 13.9

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

5. Applied rewrites 13.9%

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

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-*.f32 12.3

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

8. Applied rewrites 12.3%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 12.3

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

10. Applied rewrites 12.3%

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

11. Final simplification 12.3%

    \[\leadsto ux \cdot \left(maxCos \cdot zi\right) \]
12. Add Preprocessing

Alternative 22: 11.9% accurate, 32.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in zi around inf

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

   1. lower-*.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 13.9

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

5. Applied rewrites 13.9%

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

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-*.f32 12.3

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

8. Applied rewrites 12.3%

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

Reproduce

herbie shell --seed 2024216 
(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)))