UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 24.5s

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(maxCos * Float32(Float32(1.0) - ux))
	t_1 = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(t_0 * ux) * ux) * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(Float32(t_1 * yi), sin(t_2), Float32(cos(t_2) * Float32(xi * t_1))))
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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot \pi\right) \cdot 2\\ t_1 := \cos t\_0\\ \mathbf{if}\;uy \cdot 2 \leq 0.049949999898672104:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot zi, maxCos \cdot ux, \mathsf{fma}\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, 2 \cdot \pi\right) \cdot uy, yi, t\_1 \cdot xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \frac{\sin t\_0}{xi}, t\_1\right) \cdot xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy PI) 2.0)) (t_1 (cos t_0)))
   (if (<= (* uy 2.0) 0.049949999898672104)
     (fma
      (* (- 1.0 ux) zi)
      (* maxCos ux)
      (fma
       (*
        (fma (* (* uy uy) -1.3333333333333333) (* (* PI PI) PI) (* 2.0 PI))
        uy)
       yi
       (* t_1 xi)))
     (* (fma yi (/ (sin t_0) xi) t_1) xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (uy * ((float) M_PI)) * 2.0f;
	float t_1 = cosf(t_0);
	float tmp;
	if ((uy * 2.0f) <= 0.049949999898672104f) {
		tmp = fmaf(((1.0f - ux) * zi), (maxCos * ux), fmaf((fmaf(((uy * uy) * -1.3333333333333333f), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (2.0f * ((float) M_PI))) * uy), yi, (t_1 * xi)));
	} else {
		tmp = fmaf(yi, (sinf(t_0) / xi), t_1) * xi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(uy * Float32(pi)) * Float32(2.0))
	t_1 = cos(t_0)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.049949999898672104))
		tmp = fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(Float32(fma(Float32(Float32(uy * uy) * Float32(-1.3333333333333333)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(2.0) * Float32(pi))) * uy), yi, Float32(t_1 * xi)));
	else
		tmp = Float32(fma(yi, Float32(sin(t_0) / xi), t_1) * xi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0499499999

   1. Initial program 99.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-cos.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lower-PI.f32 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 98.9%

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

   if 0.0499499999 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.6%

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

   3. Taylor expanded in xi around inf

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 95.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.049949999898672104:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot zi, maxCos \cdot ux, \mathsf{fma}\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, 2 \cdot \pi\right) \cdot uy, yi, \cos \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \frac{\sin \left(\left(uy \cdot \pi\right) \cdot 2\right)}{xi}, \cos \left(\left(uy \cdot \pi\right) \cdot 2\right)\right) \cdot xi\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(uy * Float32(pi)) * Float32(2.0))
	return fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(sin(t_0), yi, Float32(cos(t_0) * xi)))
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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Alternative 5: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot \pi\right) \cdot 2\\ t_1 := \cos t\_0 \cdot xi\\ \mathbf{if}\;uy \cdot 2 \leq 0.049949999898672104:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot zi, maxCos \cdot ux, \mathsf{fma}\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, 2 \cdot \pi\right) \cdot uy, yi, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin t\_0, yi, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy PI) 2.0)) (t_1 (* (cos t_0) xi)))
   (if (<= (* uy 2.0) 0.049949999898672104)
     (fma
      (* (- 1.0 ux) zi)
      (* maxCos ux)
      (fma
       (*
        (fma (* (* uy uy) -1.3333333333333333) (* (* PI PI) PI) (* 2.0 PI))
        uy)
       yi
       t_1))
     (fma (sin t_0) yi t_1))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (uy * ((float) M_PI)) * 2.0f;
	float t_1 = cosf(t_0) * xi;
	float tmp;
	if ((uy * 2.0f) <= 0.049949999898672104f) {
		tmp = fmaf(((1.0f - ux) * zi), (maxCos * ux), fmaf((fmaf(((uy * uy) * -1.3333333333333333f), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (2.0f * ((float) M_PI))) * uy), yi, t_1));
	} else {
		tmp = fmaf(sinf(t_0), yi, t_1);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(uy * Float32(pi)) * Float32(2.0))
	t_1 = Float32(cos(t_0) * xi)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.049949999898672104))
		tmp = fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(Float32(fma(Float32(Float32(uy * uy) * Float32(-1.3333333333333333)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(2.0) * Float32(pi))) * uy), yi, t_1));
	else
		tmp = fma(sin(t_0), yi, t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0499499999

   1. Initial program 99.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-cos.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lower-PI.f32 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 98.9%

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

   if 0.0499499999 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.6%

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

   3. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-cos.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lower-PI.f32 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.049949999898672104:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot zi, maxCos \cdot ux, \mathsf{fma}\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, 2 \cdot \pi\right) \cdot uy, yi, \cos \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\left(uy \cdot \pi\right) \cdot 2\right), yi, \cos \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot xi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 11.1

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

5. Applied rewrites 11.1%

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

6. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-sin.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

8. Applied rewrites 97.1%

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

9. Final simplification 97.1%

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

Alternative 7: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in ux around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

5. Applied rewrites 97.1%

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

6. Final simplification 97.1%

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

Alternative 8: 93.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(1 - ux\right) \cdot zi, maxCos \cdot ux, \mathsf{fma}\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, 2 \cdot \pi\right) \cdot uy, yi, \cos \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot xi\right)\right) \end{array} \]

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(((1.0f - ux) * zi), (maxCos * ux), fmaf((fmaf(((uy * uy) * -1.3333333333333333f), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (2.0f * ((float) M_PI))) * uy), yi, (cosf(((uy * ((float) M_PI)) * 2.0f)) * xi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(Float32(fma(Float32(Float32(uy * uy) * Float32(-1.3333333333333333)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(2.0) * Float32(pi))) * uy), yi, Float32(cos(Float32(Float32(uy * Float32(pi)) * Float32(2.0))) * xi)))
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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 94.0%

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

11. Final simplification 94.0%

    \[\leadsto \mathsf{fma}\left(\left(1 - ux\right) \cdot zi, maxCos \cdot ux, \mathsf{fma}\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, 2 \cdot \pi\right) \cdot uy, yi, \cos \left(\left(uy \cdot \pi\right) \cdot 2\right) \cdot xi\right)\right) \]
12. Add Preprocessing

Alternative 9: 89.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(uy * Float32(pi)) * Float32(2.0))
	return fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(t_0, yi, Float32(cos(t_0) * xi)))
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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 90.3%

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

11. Final simplification 90.3%

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

Alternative 10: 89.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(1 - ux\right) \cdot zi, maxCos \cdot ux, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy, -1.3333333333333333, -2 \cdot \left(\left(\pi \cdot \pi\right) \cdot xi\right)\right), uy, \left(\pi \cdot yi\right) \cdot 2\right), uy, xi\right)\right) \end{array} \]

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(((1.0f - ux) * zi), (maxCos * ux), fmaf(fmaf(fmaf(((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * yi) * uy), -1.3333333333333333f, (-2.0f * ((((float) M_PI) * ((float) M_PI)) * xi))), uy, ((((float) M_PI) * yi) * 2.0f)), uy, xi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(fma(fma(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * yi) * uy), Float32(-1.3333333333333333), Float32(Float32(-2.0) * Float32(Float32(Float32(pi) * Float32(pi)) * xi))), uy, Float32(Float32(Float32(pi) * yi) * Float32(2.0))), uy, xi))
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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 89.5%

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

11. Final simplification 89.5%

    \[\leadsto \mathsf{fma}\left(\left(1 - ux\right) \cdot zi, maxCos \cdot ux, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot yi\right) \cdot uy, -1.3333333333333333, -2 \cdot \left(\left(\pi \cdot \pi\right) \cdot xi\right)\right), uy, \left(\pi \cdot yi\right) \cdot 2\right), uy, xi\right)\right) \]
12. Add Preprocessing

Alternative 11: 85.3% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(fma(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * xi) * uy), Float32(-2.0), Float32(Float32(Float32(pi) * yi) * Float32(2.0))), uy, xi))
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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 86.2%

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

11. Final simplification 86.2%

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

Alternative 12: 81.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(maxCos * ux), fma(Float32(Float32(Float32(pi) * yi) * uy), Float32(2.0), xi))
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
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 82.0%

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

11. Final simplification 82.0%

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

Alternative 13: 81.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 82.0%

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

9. Final simplification 82.0%

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

Alternative 14: 78.6% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 80.7%

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

9. Final simplification 80.7%

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

Alternative 15: 73.6% accurate, 20.8× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 77.1%

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

Alternative 16: 73.6% accurate, 20.8× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 77.1%

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

10. Applied rewrites 77.1%

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

Alternative 17: 73.7% accurate, 20.8× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 77.1%

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

10. Applied rewrites 77.1%

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

Alternative 18: 32.3% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 77.1%

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

9. Taylor expanded in yi around inf

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

11. Applied rewrites 36.0%

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

12. Final simplification 36.0%

    \[\leadsto \left(\left(\pi \cdot yi\right) \cdot uy\right) \cdot 2 \]
13. Add Preprocessing

Alternative 19: 12.1% accurate, 32.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 11.1

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

5. Applied rewrites 11.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 10.5%

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

9. Final simplification 10.5%

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

Reproduce

herbie shell --seed 2024235 
(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)))