UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 9.6s

Alternatives: 19

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

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

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

Derivation

1. Initial program 98.9%

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

Alternative 2: 98.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

4. Applied rewrites 98.8%

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

Alternative 3: 96.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathbf{if}\;uy \leq 0.023000000044703484:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, 1 - ux, \frac{\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right)\right)\right)}{zi}\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (if (<= uy 0.023000000044703484)
     (*
      (fma
       (* maxCos ux)
       (- 1.0 ux)
       (/
        (*
         (sqrt
          (-
           1.0
           (* (* (* (- 1.0 ux) (- 1.0 ux)) (* ux ux)) (* maxCos maxCos))))
         (+
          xi
          (*
           uy
           (fma
            2.0
            (* yi PI)
            (*
             uy
             (fma
              -2.0
              (* xi (* PI PI))
              (* -1.3333333333333333 (* uy (* yi (* (* PI PI) PI))))))))))
        zi))
      zi)
     (fma (cos t_0) xi (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float tmp;
	if (uy <= 0.023000000044703484f) {
		tmp = fmaf((maxCos * ux), (1.0f - ux), ((sqrtf((1.0f - ((((1.0f - ux) * (1.0f - ux)) * (ux * ux)) * (maxCos * maxCos)))) * (xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))))))))))) / zi)) * zi;
	} else {
		tmp = fmaf(cosf(t_0), xi, (sinf(t_0) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	tmp = Float32(0.0)
	if (uy <= Float32(0.023000000044703484))
		tmp = Float32(fma(Float32(maxCos * ux), Float32(Float32(1.0) - ux), Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)) * Float32(ux * ux)) * Float32(maxCos * maxCos)))) * Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))))))))))) / zi)) * zi);
	else
		tmp = fma(cos(t_0), xi, Float32(sin(t_0) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.023

   1. Initial program 98.9%

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

   2. Taylor expanded in zi around inf

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-fma.f32 N/A

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

   7. Applied rewrites 88.8%

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

   if 0.023 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

Alternative 4: 96.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathbf{if}\;uy \leq 0.023000000044703484:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, 1 - ux, \frac{\sqrt{1 - \left(\left(1 + -2 \cdot ux\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right)\right)\right)}{zi}\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (if (<= uy 0.023000000044703484)
     (*
      (fma
       (* maxCos ux)
       (- 1.0 ux)
       (/
        (*
         (sqrt (- 1.0 (* (* (+ 1.0 (* -2.0 ux)) (* ux ux)) (* maxCos maxCos))))
         (+
          xi
          (*
           uy
           (fma
            2.0
            (* yi PI)
            (*
             uy
             (fma
              -2.0
              (* xi (* PI PI))
              (* -1.3333333333333333 (* uy (* yi (* (* PI PI) PI))))))))))
        zi))
      zi)
     (fma (cos t_0) xi (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float tmp;
	if (uy <= 0.023000000044703484f) {
		tmp = fmaf((maxCos * ux), (1.0f - ux), ((sqrtf((1.0f - (((1.0f + (-2.0f * ux)) * (ux * ux)) * (maxCos * maxCos)))) * (xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))))))))))) / zi)) * zi;
	} else {
		tmp = fmaf(cosf(t_0), xi, (sinf(t_0) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	tmp = Float32(0.0)
	if (uy <= Float32(0.023000000044703484))
		tmp = Float32(fma(Float32(maxCos * ux), Float32(Float32(1.0) - ux), Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-2.0) * ux)) * Float32(ux * ux)) * Float32(maxCos * maxCos)))) * Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))))))))))) / zi)) * zi);
	else
		tmp = fma(cos(t_0), xi, Float32(sin(t_0) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.023

   1. Initial program 98.9%

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

   2. Taylor expanded in zi around inf

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-fma.f32 N/A

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

   7. Applied rewrites 88.8%

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

   8. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 88.8

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

   10. Applied rewrites 88.8%

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

   if 0.023 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

Alternative 5: 95.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

4. Applied rewrites 95.9%

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

Alternative 6: 91.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00022000000171829015:\\ \;\;\;\;\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right) + \frac{\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)}{zi}\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00022000000171829015)
   (*
    (+
     (* (* maxCos ux) (- 1.0 ux))
     (/
      (*
       (sqrt
        (- 1.0 (* (* (* (- 1.0 ux) (- 1.0 ux)) (* ux ux)) (* maxCos maxCos))))
       (+ xi (* uy (fma 2.0 (* yi PI) (* uy (* -2.0 (* xi (* PI PI))))))))
      zi))
    zi)
   (fma
    (cos (* PI (+ uy uy)))
    xi
    (*
     uy
     (fma
      -1.3333333333333333
      (* (* uy uy) (* yi (* (* PI PI) PI)))
      (* 2.0 (* yi PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00022000000171829015f) {
		tmp = (((maxCos * ux) * (1.0f - ux)) + ((sqrtf((1.0f - ((((1.0f - ux) * (1.0f - ux)) * (ux * ux)) * (maxCos * maxCos)))) * (xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * (-2.0f * (xi * (((float) M_PI) * ((float) M_PI))))))))) / zi)) * zi;
	} else {
		tmp = fmaf(cosf((((float) M_PI) * (uy + uy))), xi, (uy * fmaf(-1.3333333333333333f, ((uy * uy) * (yi * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00022000000171829015))
		tmp = Float32(Float32(Float32(Float32(maxCos * ux) * Float32(Float32(1.0) - ux)) + Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)) * Float32(ux * ux)) * Float32(maxCos * maxCos)))) * Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi))))))))) / zi)) * zi);
	else
		tmp = fma(cos(Float32(Float32(pi) * Float32(uy + uy))), xi, Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(yi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 2.20000002e-4

   1. Initial program 98.9%

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

   2. Taylor expanded in zi around inf

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-fma.f32 N/A

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

   7. Applied rewrites 88.8%

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

   8. Taylor expanded in xi around inf

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 85.3

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

   10. Applied rewrites 85.3%

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

   12. Applied rewrites 85.3%

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

   if 2.20000002e-4 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. pow3 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. lift-PI.f32 85.7

          \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 85.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  maxCos
  (* ux zi)
  (fma
   xi
   (sin (fma 0.5 PI (* 2.0 (* uy PI))))
   (*
    yi
    (*
     uy
     (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * zi), fmaf(xi, sinf(fmaf(0.5f, ((float) M_PI), (2.0f * (uy * ((float) M_PI))))), (yi * (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * zi), fma(xi, sin(fma(Float32(0.5), Float32(pi), Float32(Float32(2.0) * Float32(uy * Float32(pi))))), Float32(yi * Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))))))
end

Derivation

1. Initial program 98.9%

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

   1. lift-cos.f32 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lift-PI.f32 98.8

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

3. Applied rewrites 98.8%

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

4. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

6. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

7. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. pow3 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. pow2 N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   11. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   12. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   13. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   15. lift-PI.f32 91.1

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right)\right) \]

9. Applied rewrites 91.1%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \sin \left(\mathsf{fma}\left(0.5, \pi, 2 \cdot \left(uy \cdot \pi\right)\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 8: 90.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\\ \mathbf{if}\;uy \leq 0.00021499999274965376:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, 1 - ux, \frac{\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(xi + t\_0\right)}{zi}\right) \cdot zi\\ \mathbf{elif}\;uy \leq 0.026000000536441803:\\ \;\;\;\;xi + yi \cdot \mathsf{fma}\left(-2, \frac{\left(uy \cdot uy\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)}{yi}, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (* yi PI)))))
   (if (<= uy 0.00021499999274965376)
     (*
      (fma
       (* maxCos ux)
       (- 1.0 ux)
       (/
        (*
         (sqrt
          (-
           1.0
           (* (* (* (- 1.0 ux) (- 1.0 ux)) (* ux ux)) (* maxCos maxCos))))
         (+ xi t_0))
        zi))
      zi)
     (if (<= uy 0.026000000536441803)
       (+
        xi
        (*
         yi
         (fma
          -2.0
          (/ (* (* uy uy) (* xi (* PI PI))) yi)
          (*
           uy
           (fma
            -1.3333333333333333
            (* (* uy uy) (* (* PI PI) PI))
            (* 2.0 PI))))))
       (fma (cos (* PI (+ uy uy))) xi t_0)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * (yi * ((float) M_PI)));
	float tmp;
	if (uy <= 0.00021499999274965376f) {
		tmp = fmaf((maxCos * ux), (1.0f - ux), ((sqrtf((1.0f - ((((1.0f - ux) * (1.0f - ux)) * (ux * ux)) * (maxCos * maxCos)))) * (xi + t_0)) / zi)) * zi;
	} else if (uy <= 0.026000000536441803f) {
		tmp = xi + (yi * fmaf(-2.0f, (((uy * uy) * (xi * (((float) M_PI) * ((float) M_PI)))) / yi), (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI))))));
	} else {
		tmp = fmaf(cosf((((float) M_PI) * (uy + uy))), xi, t_0);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))
	tmp = Float32(0.0)
	if (uy <= Float32(0.00021499999274965376))
		tmp = Float32(fma(Float32(maxCos * ux), Float32(Float32(1.0) - ux), Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)) * Float32(ux * ux)) * Float32(maxCos * maxCos)))) * Float32(xi + t_0)) / zi)) * zi);
	elseif (uy <= Float32(0.026000000536441803))
		tmp = Float32(xi + Float32(yi * fma(Float32(-2.0), Float32(Float32(Float32(uy * uy) * Float32(xi * Float32(Float32(pi) * Float32(pi)))) / yi), Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))))));
	else
		tmp = fma(cos(Float32(Float32(pi) * Float32(uy + uy))), xi, t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if uy < 2.14999993e-4

   1. Initial program 98.9%

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

   2. Taylor expanded in zi around inf

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 81.3

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

   7. Applied rewrites 81.3%

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

   if 2.14999993e-4 < uy < 0.0260000005

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   7. Applied rewrites 81.2%

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

   8. Taylor expanded in yi around inf

      \[\leadsto xi + yi \cdot \left(-2 \cdot \frac{{uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{yi} + \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

         \[\leadsto xi + yi \cdot \mathsf{fma}\left(-2, \frac{{uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{yi}, uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. Applied rewrites 81.2%

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

   if 0.0260000005 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 82.1

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

   7. Applied rewrites 82.1%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00022000000171829015:\\ \;\;\;\;\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right) + \frac{\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)}{zi}\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00022000000171829015)
   (*
    (+
     (* (* maxCos ux) (- 1.0 ux))
     (/
      (*
       (sqrt
        (- 1.0 (* (* (* (- 1.0 ux) (- 1.0 ux)) (* ux ux)) (* maxCos maxCos))))
       (+ xi (* uy (fma 2.0 (* yi PI) (* uy (* -2.0 (* xi (* PI PI))))))))
      zi))
    zi)
   (fma
    (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))
    xi
    (* (sin (* PI (+ uy uy))) yi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00022000000171829015f) {
		tmp = (((maxCos * ux) * (1.0f - ux)) + ((sqrtf((1.0f - ((((1.0f - ux) * (1.0f - ux)) * (ux * ux)) * (maxCos * maxCos)))) * (xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * (-2.0f * (xi * (((float) M_PI) * ((float) M_PI))))))))) / zi)) * zi;
	} else {
		tmp = fmaf((1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI))))), xi, (sinf((((float) M_PI) * (uy + uy))) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00022000000171829015))
		tmp = Float32(Float32(Float32(Float32(maxCos * ux) * Float32(Float32(1.0) - ux)) + Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)) * Float32(ux * ux)) * Float32(maxCos * maxCos)))) * Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi))))))))) / zi)) * zi);
	else
		tmp = fma(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))), xi, Float32(sin(Float32(Float32(pi) * Float32(uy + uy))) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 2.20000002e-4

   1. Initial program 98.9%

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

   2. Taylor expanded in zi around inf

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-fma.f32 N/A

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

   7. Applied rewrites 88.8%

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

   8. Taylor expanded in xi around inf

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 85.3

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

   10. Applied rewrites 85.3%

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

   12. Applied rewrites 85.3%

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

   if 2.20000002e-4 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 84.7

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

   7. Applied rewrites 84.7%

      \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 89.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00022000000171829015:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, 1 - ux, \frac{\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)}{zi}\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00022000000171829015)
   (*
    (fma
     (* maxCos ux)
     (- 1.0 ux)
     (/
      (*
       (sqrt
        (- 1.0 (* (* (* (- 1.0 ux) (- 1.0 ux)) (* ux ux)) (* maxCos maxCos))))
       (+ xi (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI))))))
      zi))
    zi)
   (fma
    (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))
    xi
    (* (sin (* PI (+ uy uy))) yi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00022000000171829015f) {
		tmp = fmaf((maxCos * ux), (1.0f - ux), ((sqrtf((1.0f - ((((1.0f - ux) * (1.0f - ux)) * (ux * ux)) * (maxCos * maxCos)))) * (xi + (uy * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI))))))) / zi)) * zi;
	} else {
		tmp = fmaf((1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI))))), xi, (sinf((((float) M_PI) * (uy + uy))) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00022000000171829015))
		tmp = Float32(fma(Float32(maxCos * ux), Float32(Float32(1.0) - ux), Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)) * Float32(ux * ux)) * Float32(maxCos * maxCos)))) * Float32(xi + Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi))))))) / zi)) * zi);
	else
		tmp = fma(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))), xi, Float32(sin(Float32(Float32(pi) * Float32(uy + uy))) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 2.20000002e-4

   1. Initial program 98.9%

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

   2. Taylor expanded in zi around inf

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 85.3

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

   7. Applied rewrites 85.3%

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

   if 2.20000002e-4 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 84.7

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

   7. Applied rewrites 84.7%

      \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 89.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00021499999274965376:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, 1 - ux, \frac{\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}{zi}\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00021499999274965376)
   (*
    (fma
     (* maxCos ux)
     (- 1.0 ux)
     (/
      (*
       (sqrt
        (- 1.0 (* (* (* (- 1.0 ux) (- 1.0 ux)) (* ux ux)) (* maxCos maxCos))))
       (+ xi (* 2.0 (* uy (* yi PI)))))
      zi))
    zi)
   (fma
    (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))
    xi
    (* (sin (* PI (+ uy uy))) yi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00021499999274965376f) {
		tmp = fmaf((maxCos * ux), (1.0f - ux), ((sqrtf((1.0f - ((((1.0f - ux) * (1.0f - ux)) * (ux * ux)) * (maxCos * maxCos)))) * (xi + (2.0f * (uy * (yi * ((float) M_PI)))))) / zi)) * zi;
	} else {
		tmp = fmaf((1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI))))), xi, (sinf((((float) M_PI) * (uy + uy))) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00021499999274965376))
		tmp = Float32(fma(Float32(maxCos * ux), Float32(Float32(1.0) - ux), Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)) * Float32(ux * ux)) * Float32(maxCos * maxCos)))) * Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))))) / zi)) * zi);
	else
		tmp = fma(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))), xi, Float32(sin(Float32(Float32(pi) * Float32(uy + uy))) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 2.14999993e-4

   1. Initial program 98.9%

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

   2. Taylor expanded in zi around inf

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 81.3

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

   7. Applied rewrites 81.3%

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

   if 2.14999993e-4 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]
   6. Step-by-step derivation

      1. sin-+PI/2-rev N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 84.7

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

   7. Applied rewrites 84.7%

      \[\leadsto \mathsf{fma}\left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), xi, \sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot yi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 84.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.026000000536441803:\\ \;\;\;\;xi + yi \cdot \mathsf{fma}\left(-2, \frac{\left(uy \cdot uy\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)}{yi}, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.026000000536441803)
   (+
    xi
    (*
     yi
     (fma
      -2.0
      (/ (* (* uy uy) (* xi (* PI PI))) yi)
      (*
       uy
       (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI))))))
   (fma (cos (* PI (+ uy uy))) xi (* 2.0 (* uy (* yi PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.026000000536441803f) {
		tmp = xi + (yi * fmaf(-2.0f, (((uy * uy) * (xi * (((float) M_PI) * ((float) M_PI)))) / yi), (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI))))));
	} else {
		tmp = fmaf(cosf((((float) M_PI) * (uy + uy))), xi, (2.0f * (uy * (yi * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.026000000536441803))
		tmp = Float32(xi + Float32(yi * fma(Float32(-2.0), Float32(Float32(Float32(uy * uy) * Float32(xi * Float32(Float32(pi) * Float32(pi)))) / yi), Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))))));
	else
		tmp = fma(cos(Float32(Float32(pi) * Float32(uy + uy))), xi, Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0260000005

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   7. Applied rewrites 81.2%

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

   8. Taylor expanded in yi around inf

      \[\leadsto xi + yi \cdot \left(-2 \cdot \frac{{uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{yi} + \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

         \[\leadsto xi + yi \cdot \mathsf{fma}\left(-2, \frac{{uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{yi}, uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. Applied rewrites 81.2%

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

   if 0.0260000005 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 82.1

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

   7. Applied rewrites 82.1%

      \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(uy + uy\right)\right), xi, 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 81.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ xi + yi \cdot \mathsf{fma}\left(-2, \frac{\left(uy \cdot uy\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)}{yi}, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  xi
  (*
   yi
   (fma
    -2.0
    (/ (* (* uy uy) (* xi (* PI PI))) yi)
    (*
     uy
     (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (yi * fmaf(-2.0f, (((uy * uy) * (xi * (((float) M_PI) * ((float) M_PI)))) / yi), (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(yi * fma(Float32(-2.0), Float32(Float32(Float32(uy * uy) * Float32(xi * Float32(Float32(pi) * Float32(pi)))) / yi), Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))))))
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

4. Applied rewrites 90.3%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   5. lift-PI.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

7. Applied rewrites 81.2%

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

8. Taylor expanded in yi around inf

   \[\leadsto xi + yi \cdot \left(-2 \cdot \frac{{uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{yi} + \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

      \[\leadsto xi + yi \cdot \mathsf{fma}\left(-2, \frac{{uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{yi}, uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

10. Applied rewrites 81.2%

    \[\leadsto xi + yi \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{\left(uy \cdot uy\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)}{yi}}, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \]
11. Add Preprocessing

Alternative 14: 81.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  xi
  (*
   uy
   (fma
    2.0
    (* yi PI)
    (*
     uy
     (fma
      -2.0
      (* xi (* PI PI))
      (* -1.3333333333333333 (* uy (* yi (* (* PI PI) PI))))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)))))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))))))))))
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

4. Applied rewrites 90.3%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   5. lift-PI.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

7. Applied rewrites 81.2%

   \[\leadsto xi + \color{blue}{uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 15: 79.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.026000000536441803:\\ \;\;\;\;xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.026000000536441803)
   (+
    xi
    (*
     uy
     (*
      yi
      (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))))
   (* 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 <= 0.026000000536441803f) {
		tmp = xi + (uy * (yi * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))));
	} 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 (uy <= Float32(0.026000000536441803))
		tmp = Float32(xi + Float32(uy * Float32(yi * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi))))));
	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 uy < 0.0260000005

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   7. Applied rewrites 81.2%

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

   8. Taylor expanded in yi around inf

      \[\leadsto xi + uy \cdot \left(yi \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

         \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. pow3 N/A

         \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      7. lift-*.f32 N/A

         \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      10. lift-*.f32 N/A

          \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      11. lift-PI.f32 N/A

          \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      13. lift-PI.f32 77.0

          \[\leadsto xi + uy \cdot \left(yi \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \]

   10. Applied rewrites 77.0%

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

   if 0.0260000005 < uy

   1. Initial program 98.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in xi around inf

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

      1. lower-*.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 52.7

         \[\leadsto xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   7. Applied rewrites 52.7%

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

Alternative 16: 77.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

4. Applied rewrites 90.3%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   5. lift-PI.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right) \]

7. Applied rewrites 81.2%

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

8. Taylor expanded in xi around inf

   \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

   2. pow2 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. lift-PI.f32 N/A

      \[\leadsto xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 77.8

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

10. Applied rewrites 77.8%

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

Alternative 17: 74.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (single(2.0) * (uy * (yi * single(pi))));
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

4. Applied rewrites 90.3%

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

5. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 74.1

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

7. Applied rewrites 74.1%

   \[\leadsto xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)} \]
8. Add Preprocessing

Alternative 18: 49.6% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    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 = xi + (maxcos * (ux * zi))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 49.6

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.6%

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

Alternative 19: 12.1% accurate, 22.3× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    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 98.9%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 49.6

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.6%

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

8. Taylor expanded in xi around 0

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

   1. lift-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lift-*.f32 12.1

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 12.1%

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

Reproduce

herbie shell --seed 2025134 
(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)))