UniformSampleCone 2

Percentage Accurate: 98.9% → 98.8%

Time: 24.0s

Alternatives: 24

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot 2\right) \cdot \pi\\ t_1 := ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\\ \left(\left(\sqrt{1 + t\_1 \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \cos t\_0\right) \cdot xi + \left(\sin t\_0 \cdot \sqrt{\left(ux \cdot \left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right)\right) \cdot \left(\mathsf{fma}\left(maxCos, maxCos \cdot -2, \frac{\mathsf{fma}\left(maxCos, -maxCos, \frac{1}{\mathsf{fma}\left(ux, ux, 0\right)}\right)}{-ux}\right) \cdot \frac{-1}{ux} - maxCos \cdot maxCos\right)}\right) \cdot yi\right) + t\_1 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy 2.0) PI)) (t_1 (* ux (* (- 1.0 ux) maxCos))))
   (+
    (+
     (* (* (sqrt (+ 1.0 (* t_1 (* ux (* maxCos (+ ux -1.0)))))) (cos t_0)) xi)
     (*
      (*
       (sin t_0)
       (sqrt
        (*
         (* ux (* ux (fma ux ux 0.0)))
         (-
          (*
           (fma
            maxCos
            (* maxCos -2.0)
            (/ (fma maxCos (- maxCos) (/ 1.0 (fma ux ux 0.0))) (- ux)))
           (/ -1.0 ux))
          (* maxCos maxCos)))))
      yi))
    (* t_1 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (uy * 2.0f) * ((float) M_PI);
	float t_1 = ux * ((1.0f - ux) * maxCos);
	return (((sqrtf((1.0f + (t_1 * (ux * (maxCos * (ux + -1.0f)))))) * cosf(t_0)) * xi) + ((sinf(t_0) * sqrtf(((ux * (ux * fmaf(ux, ux, 0.0f))) * ((fmaf(maxCos, (maxCos * -2.0f), (fmaf(maxCos, -maxCos, (1.0f / fmaf(ux, ux, 0.0f))) / -ux)) * (-1.0f / ux)) - (maxCos * maxCos))))) * yi)) + (t_1 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	t_1 = Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos))
	return Float32(Float32(Float32(Float32(sqrt(Float32(Float32(1.0) + Float32(t_1 * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0))))))) * cos(t_0)) * xi) + Float32(Float32(sin(t_0) * sqrt(Float32(Float32(ux * Float32(ux * fma(ux, ux, Float32(0.0)))) * Float32(Float32(fma(maxCos, Float32(maxCos * Float32(-2.0)), Float32(fma(maxCos, Float32(-maxCos), Float32(Float32(1.0) / fma(ux, ux, Float32(0.0)))) / Float32(-ux))) * Float32(Float32(-1.0) / ux)) - Float32(maxCos * maxCos))))) * yi)) + Float32(t_1 * zi))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in ux around inf

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

   1. *-lowering-*.f32 N/A

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

   2. pow-lowering-pow.f32 N/A

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

   3. --lowering--.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. pow-lowering-pow.f32 N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. accelerator-lowering-fma.f32 N/A

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

   13. /-lowering-/.f32 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f32 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f32 98.9

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

5. Simplified 98.9%

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

6. Taylor expanded in ux around -inf

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

8. Simplified 98.9%

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. *-lowering-*.f32 N/A

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

   4. cube-mult N/A

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

   5. *-lowering-*.f32 N/A

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

   6. +-lft-identity N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(ux \cdot \color{blue}{\left(0 + ux \cdot ux\right)}\right) \cdot ux\right) \cdot \left(\left(\mathsf{neg}\left(maxCos \cdot maxCos\right)\right) - \frac{\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), maxCos, \frac{1}{ux \cdot ux}\right)}{ux}}{ux}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(ux \cdot \color{blue}{\left(ux \cdot ux + 0\right)}\right) \cdot ux\right) \cdot \left(\left(\mathsf{neg}\left(maxCos \cdot maxCos\right)\right) - \frac{\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), maxCos, \frac{1}{ux \cdot ux}\right)}{ux}}{ux}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. accelerator-lowering-fma.f32 98.9

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot \color{blue}{\mathsf{fma}\left(ux, ux, 0\right)}\right) \cdot ux\right) \cdot \left(\left(-maxCos \cdot maxCos\right) - \frac{\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\mathsf{fma}\left(-maxCos, maxCos, \frac{1}{ux \cdot ux}\right)}{ux}}{ux}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

10. Applied egg-rr 98.9%

    \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right) \cdot ux\right)} \cdot \left(\left(-maxCos \cdot maxCos\right) - \frac{\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\mathsf{fma}\left(-maxCos, maxCos, \frac{1}{ux \cdot ux}\right)}{ux}}{ux}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
11. Step-by-step derivation

    1. div-inv N/A

       \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right) \cdot ux\right) \cdot \left(\left(\mathsf{neg}\left(maxCos \cdot maxCos\right)\right) - \color{blue}{\left(\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\left(\mathsf{neg}\left(maxCos\right)\right) \cdot maxCos + \frac{1}{ux \cdot ux}}{ux}\right) \cdot \frac{1}{ux}}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right) \cdot ux\right) \cdot \left(\left(\mathsf{neg}\left(maxCos \cdot maxCos\right)\right) - \color{blue}{\left(\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\left(\mathsf{neg}\left(maxCos\right)\right) \cdot maxCos + \frac{1}{ux \cdot ux}}{ux}\right) \cdot \frac{1}{ux}}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

12. Applied egg-rr 98.9%

    \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right) \cdot ux\right) \cdot \left(\left(-maxCos \cdot maxCos\right) - \color{blue}{\mathsf{fma}\left(maxCos, maxCos \cdot -2, 0 - \frac{\mathsf{fma}\left(maxCos, -maxCos, \frac{1}{\mathsf{fma}\left(ux, ux, 0\right)}\right)}{ux}\right) \cdot \frac{1}{ux}}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

13. Final simplification 98.9%

    \[\leadsto \left(\left(\sqrt{1 + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right)\right) \cdot \left(\mathsf{fma}\left(maxCos, maxCos \cdot -2, \frac{\mathsf{fma}\left(maxCos, -maxCos, \frac{1}{\mathsf{fma}\left(ux, ux, 0\right)}\right)}{-ux}\right) \cdot \frac{-1}{ux} - maxCos \cdot maxCos\right)}\right) \cdot yi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi \]
14. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\\ t_1 := \left(uy \cdot 2\right) \cdot \pi\\ t\_0 \cdot zi + \left(\left(\sqrt{1 + t\_0 \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \cos t\_1\right) \cdot xi + yi \cdot \left(\sin t\_1 \cdot \sqrt{\left(ux \cdot \left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right)\right) \cdot \left(\frac{\frac{\mathsf{fma}\left(-maxCos, maxCos, \frac{1}{ux \cdot ux}\right)}{ux} - \left(maxCos \cdot maxCos\right) \cdot -2}{ux} - maxCos \cdot maxCos\right)}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (* (- 1.0 ux) maxCos))) (t_1 (* (* uy 2.0) PI)))
   (+
    (* t_0 zi)
    (+
     (* (* (sqrt (+ 1.0 (* t_0 (* ux (* maxCos (+ ux -1.0)))))) (cos t_1)) xi)
     (*
      yi
      (*
       (sin t_1)
       (sqrt
        (*
         (* ux (* ux (fma ux ux 0.0)))
         (-
          (/
           (-
            (/ (fma (- maxCos) maxCos (/ 1.0 (* ux ux))) ux)
            (* (* maxCos maxCos) -2.0))
           ux)
          (* maxCos maxCos))))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * ((1.0f - ux) * maxCos);
	float t_1 = (uy * 2.0f) * ((float) M_PI);
	return (t_0 * zi) + (((sqrtf((1.0f + (t_0 * (ux * (maxCos * (ux + -1.0f)))))) * cosf(t_1)) * xi) + (yi * (sinf(t_1) * sqrtf(((ux * (ux * fmaf(ux, ux, 0.0f))) * ((((fmaf(-maxCos, maxCos, (1.0f / (ux * ux))) / ux) - ((maxCos * maxCos) * -2.0f)) / ux) - (maxCos * maxCos)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos))
	t_1 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(t_0 * zi) + Float32(Float32(Float32(sqrt(Float32(Float32(1.0) + Float32(t_0 * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0))))))) * cos(t_1)) * xi) + Float32(yi * Float32(sin(t_1) * sqrt(Float32(Float32(ux * Float32(ux * fma(ux, ux, Float32(0.0)))) * Float32(Float32(Float32(Float32(fma(Float32(-maxCos), maxCos, Float32(Float32(1.0) / Float32(ux * ux))) / ux) - Float32(Float32(maxCos * maxCos) * Float32(-2.0))) / ux) - Float32(maxCos * maxCos))))))))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in ux around inf

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

   1. *-lowering-*.f32 N/A

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

   2. pow-lowering-pow.f32 N/A

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

   3. --lowering--.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. pow-lowering-pow.f32 N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. /-lowering-/.f32 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. accelerator-lowering-fma.f32 N/A

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

   13. /-lowering-/.f32 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f32 N/A

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

   16. unpow2 N/A

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

   17. *-lowering-*.f32 98.9

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

5. Simplified 98.9%

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

6. Taylor expanded in ux around -inf

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

8. Simplified 98.9%

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. *-lowering-*.f32 N/A

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

   4. cube-mult N/A

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

   5. *-lowering-*.f32 N/A

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

   6. +-lft-identity N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(ux \cdot \color{blue}{\left(0 + ux \cdot ux\right)}\right) \cdot ux\right) \cdot \left(\left(\mathsf{neg}\left(maxCos \cdot maxCos\right)\right) - \frac{\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), maxCos, \frac{1}{ux \cdot ux}\right)}{ux}}{ux}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(ux \cdot \color{blue}{\left(ux \cdot ux + 0\right)}\right) \cdot ux\right) \cdot \left(\left(\mathsf{neg}\left(maxCos \cdot maxCos\right)\right) - \frac{\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), maxCos, \frac{1}{ux \cdot ux}\right)}{ux}}{ux}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. accelerator-lowering-fma.f32 98.9

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot \color{blue}{\mathsf{fma}\left(ux, ux, 0\right)}\right) \cdot ux\right) \cdot \left(\left(-maxCos \cdot maxCos\right) - \frac{\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\mathsf{fma}\left(-maxCos, maxCos, \frac{1}{ux \cdot ux}\right)}{ux}}{ux}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

10. Applied egg-rr 98.9%

    \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right) \cdot ux\right)} \cdot \left(\left(-maxCos \cdot maxCos\right) - \frac{\left(maxCos \cdot maxCos\right) \cdot -2 - \frac{\mathsf{fma}\left(-maxCos, maxCos, \frac{1}{ux \cdot ux}\right)}{ux}}{ux}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

11. Final simplification 98.9%

    \[\leadsto \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \left(\left(\sqrt{1 + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \cdot xi + yi \cdot \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(ux \cdot \mathsf{fma}\left(ux, ux, 0\right)\right)\right) \cdot \left(\frac{\frac{\mathsf{fma}\left(-maxCos, maxCos, \frac{1}{ux \cdot ux}\right)}{ux} - \left(maxCos \cdot maxCos\right) \cdot -2}{ux} - maxCos \cdot maxCos\right)}\right)\right) \]
12. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot xi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)}, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right)}}} \]

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

4. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right), zi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)}, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi, \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\right)\right)} \]

5. Final simplification 98.9%

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

Alternative 5: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in ux around 0

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

   1. *-lowering-*.f32 N/A

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

   2. sin-lowering-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. PI-lowering-PI.f32 98.7

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 6: 97.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.07999999821186066:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(t\_1, \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right), \left(t\_1 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(uy \cdot -1.3333333333333333\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI)))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.07999999821186066)
     (fma
      maxCos
      (* ux (* (- 1.0 ux) zi))
      (fma
       t_1
       (fma (* PI (* 2.0 yi)) uy xi)
       (*
        (*
         t_1
         (fma
          -2.0
          (* xi (* PI PI))
          (* (* uy -1.3333333333333333) (* yi (* PI (* PI PI))))))
        (* uy uy))))
     (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.07999999821186066f) {
		tmp = fmaf(maxCos, (ux * ((1.0f - ux) * zi)), fmaf(t_1, fmaf((((float) M_PI) * (2.0f * yi)), uy, xi), ((t_1 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), ((uy * -1.3333333333333333f) * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))))) * (uy * uy))));
	} else {
		tmp = fmaf(xi, cosf(t_0), (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	t_1 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.07999999821186066))
		tmp = fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), fma(t_1, fma(Float32(Float32(pi) * Float32(Float32(2.0) * yi)), uy, xi), Float32(Float32(t_1 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(uy * Float32(-1.3333333333333333)) * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))) * Float32(uy * uy))));
	else
		tmp = fma(xi, cos(t_0), Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0799999982

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

   5. Simplified 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(-1.3333333333333333 \cdot uy\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)} \]

   if 0.0799999982 < (*.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. accelerator-lowering-fma.f32 N/A

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

      2. cos-lowering-cos.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f32 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. PI-lowering-PI.f32 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. sin-lowering-sin.f32 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f32 N/A

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

      13. *-lowering-*.f32 N/A

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

      14. PI-lowering-PI.f32 90.5

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

   5. Simplified 90.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.07999999821186066:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(uy \cdot -1.3333333333333333\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right), yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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

   2. associate-+l+ N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. cos-lowering-cos.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. PI-lowering-PI.f32 N/A

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

   10. accelerator-lowering-fma.f32 N/A

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

   11. sin-lowering-sin.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-lowering-*.f32 N/A

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

   15. *-lowering-*.f32 N/A

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

   16. PI-lowering-PI.f32 N/A

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

   17. *-lowering-*.f32 N/A

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

   18. *-lowering-*.f32 95.5

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

5. Simplified 95.5%

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

6. Final simplification 95.5%

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

Alternative 8: 92.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.09000000357627869:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(t\_0, \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right), \left(t\_0 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(uy \cdot -1.3333333333333333\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(xi, \frac{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}{zi}, 2 \cdot \frac{uy \cdot \left(\pi \cdot yi\right)}{zi}\right), maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.09000000357627869)
     (fma
      maxCos
      (* ux (* (- 1.0 ux) zi))
      (fma
       t_0
       (fma (* PI (* 2.0 yi)) uy xi)
       (*
        (*
         t_0
         (fma
          -2.0
          (* xi (* PI PI))
          (* (* uy -1.3333333333333333) (* yi (* PI (* PI PI))))))
        (* uy uy))))
     (*
      zi
      (fma
       t_0
       (fma xi (/ (cos (* (* uy 2.0) PI)) zi) (* 2.0 (/ (* uy (* PI yi)) zi)))
       (* maxCos (* ux (- 1.0 ux))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.09000000357627869f) {
		tmp = fmaf(maxCos, (ux * ((1.0f - ux) * zi)), fmaf(t_0, fmaf((((float) M_PI) * (2.0f * yi)), uy, xi), ((t_0 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), ((uy * -1.3333333333333333f) * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))))) * (uy * uy))));
	} else {
		tmp = zi * fmaf(t_0, fmaf(xi, (cosf(((uy * 2.0f) * ((float) M_PI))) / zi), (2.0f * ((uy * (((float) M_PI) * yi)) / zi))), (maxCos * (ux * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.09000000357627869))
		tmp = fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), fma(t_0, fma(Float32(Float32(pi) * Float32(Float32(2.0) * yi)), uy, xi), Float32(Float32(t_0 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(uy * Float32(-1.3333333333333333)) * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))) * Float32(uy * uy))));
	else
		tmp = Float32(zi * fma(t_0, fma(xi, Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) / zi), Float32(Float32(2.0) * Float32(Float32(uy * Float32(Float32(pi) * yi)) / zi))), Float32(maxCos * Float32(ux * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0900000036

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

   5. Simplified 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(-1.3333333333333333 \cdot uy\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)} \]

   if 0.0900000036 < (*.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

   4. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot xi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)}, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right)}}} \]

   5. Taylor expanded in zi around inf

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

      1. *-lowering-*.f32 N/A

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

      2. +-commutative N/A

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

   7. Simplified 97.2%

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

   8. Taylor expanded in uy around 0

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

      1. *-lowering-*.f32 N/A

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

      2. /-lowering-/.f32 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. PI-lowering-PI.f32 63.6

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

   10. Simplified 63.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.09000000357627869:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(uy \cdot -1.3333333333333333\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(xi, \frac{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}{zi}, 2 \cdot \frac{uy \cdot \left(\pi \cdot yi\right)}{zi}\right), maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.1809999942779541:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \mathsf{fma}\left(xi, t\_0, uy \cdot \left(t\_0 \cdot \mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(uy \cdot xi\right)\right), \pi \cdot \left(2 \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot t\_0, ux \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.1809999942779541)
     (+
      (* (* ux (* (- 1.0 ux) maxCos)) zi)
      (fma
       xi
       t_0
       (* uy (* t_0 (fma -2.0 (* PI (* PI (* uy xi))) (* PI (* 2.0 yi)))))))
     (fma
      xi
      (* (cos (* uy (* 2.0 PI))) t_0)
      (* ux (* maxCos (* (- 1.0 ux) zi)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.1809999942779541f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + fmaf(xi, t_0, (uy * (t_0 * fmaf(-2.0f, (((float) M_PI) * (((float) M_PI) * (uy * xi))), (((float) M_PI) * (2.0f * yi))))));
	} else {
		tmp = fmaf(xi, (cosf((uy * (2.0f * ((float) M_PI)))) * t_0), (ux * (maxCos * ((1.0f - ux) * zi))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.1809999942779541))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + fma(xi, t_0, Float32(uy * Float32(t_0 * fma(Float32(-2.0), Float32(Float32(pi) * Float32(Float32(pi) * Float32(uy * xi))), Float32(Float32(pi) * Float32(Float32(2.0) * yi)))))));
	else
		tmp = fma(xi, Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * t_0), Float32(ux * Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * zi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.18099999

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

   5. Simplified 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, \left(\left(uy \cdot xi\right) \cdot \pi\right) \cdot \pi, \left(2 \cdot yi\right) \cdot \pi\right)\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.18099999 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.1%

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

   3. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \cos \left(uy \cdot \left(\pi \cdot 2\right)\right), ux \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.1809999942779541:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(uy \cdot xi\right)\right), \pi \cdot \left(2 \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, ux \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.18000000715255737:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \mathsf{fma}\left(xi, t\_0, uy \cdot \left(t\_0 \cdot \mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(uy \cdot xi\right)\right), \pi \cdot \left(2 \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, t\_0 \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.18000000715255737)
     (+
      (* (* ux (* (- 1.0 ux) maxCos)) zi)
      (fma
       xi
       t_0
       (* uy (* t_0 (fma -2.0 (* PI (* PI (* uy xi))) (* PI (* 2.0 yi)))))))
     (fma
      xi
      (* t_0 (cos (* (* uy 2.0) PI)))
      (* maxCos (* ux (* (- 1.0 ux) zi)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.18000000715255737f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + fmaf(xi, t_0, (uy * (t_0 * fmaf(-2.0f, (((float) M_PI) * (((float) M_PI) * (uy * xi))), (((float) M_PI) * (2.0f * yi))))));
	} else {
		tmp = fmaf(xi, (t_0 * cosf(((uy * 2.0f) * ((float) M_PI)))), (maxCos * (ux * ((1.0f - ux) * zi))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.18000000715255737))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + fma(xi, t_0, Float32(uy * Float32(t_0 * fma(Float32(-2.0), Float32(Float32(pi) * Float32(Float32(pi) * Float32(uy * xi))), Float32(Float32(pi) * Float32(Float32(2.0) * yi)))))));
	else
		tmp = fma(xi, Float32(t_0 * cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))), Float32(maxCos * Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.180000007

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

   5. Simplified 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, \left(\left(uy \cdot xi\right) \cdot \pi\right) \cdot \pi, \left(2 \cdot yi\right) \cdot \pi\right)\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.180000007 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.2%

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

   4. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot xi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)}, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right)}}} \]

   5. Taylor expanded in yi around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

   7. Simplified 58.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.18000000715255737:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(uy \cdot xi\right)\right), \pi \cdot \left(2 \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (fma
    maxCos
    (* ux (* (- 1.0 ux) zi))
    (fma
     t_0
     (fma (* PI (* 2.0 yi)) uy xi)
     (*
      (*
       t_0
       (fma
        -2.0
        (* xi (* PI PI))
        (* (* uy -1.3333333333333333) (* yi (* PI (* PI PI))))))
      (* uy uy))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	return fmaf(maxCos, (ux * ((1.0f - ux) * zi)), fmaf(t_0, fmaf((((float) M_PI) * (2.0f * yi)), uy, xi), ((t_0 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), ((uy * -1.3333333333333333f) * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))))) * (uy * uy))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	return fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), fma(t_0, fma(Float32(Float32(pi) * Float32(Float32(2.0) * yi)), uy, xi), Float32(Float32(t_0 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(uy * Float32(-1.3333333333333333)) * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))) * Float32(uy * uy))))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

5. Simplified 88.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right), \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), \left(-1.3333333333333333 \cdot uy\right) \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)} \]

6. Final simplification 88.2%

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

Alternative 12: 85.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

5. Simplified 84.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, uy \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, \left(\left(uy \cdot xi\right) \cdot \pi\right) \cdot \pi, \left(2 \cdot yi\right) \cdot \pi\right)\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

6. Final simplification 84.1%

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

Alternative 13: 85.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

5. Simplified 84.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(-2, \left(\left(uy \cdot xi\right) \cdot \pi\right) \cdot \pi, \left(2 \cdot yi\right) \cdot \pi\right), ux \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)\right)} \]

6. Final simplification 84.0%

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

Alternative 14: 83.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.007000000216066837:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.007000000216066837)
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) zi)
    (*
     (sqrt
      (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
     (fma (* PI (* 2.0 yi)) uy xi)))
   (fma yi (sin (* 2.0 (* uy PI))) (* (* ux maxCos) (* (- 1.0 ux) zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.007000000216066837f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf((((float) M_PI) * (2.0f * yi)), uy, xi));
	} else {
		tmp = fmaf(yi, sinf((2.0f * (uy * ((float) M_PI)))), ((ux * maxCos) * ((1.0f - ux) * zi)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.007000000216066837))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(Float32(Float32(pi) * Float32(Float32(2.0) * yi)), uy, xi)));
	else
		tmp = fma(yi, sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(Float32(ux * maxCos) * Float32(Float32(Float32(1.0) - ux) * zi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00700000022

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. *-lowering-*.f32 N/A

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

   5. Simplified 96.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.00700000022 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 98.0%

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

   3. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

   5. Simplified 55.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)}, yi \cdot \sin \left(uy \cdot \left(\pi \cdot 2\right)\right), ux \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)} \]

   6. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. sin-lowering-sin.f32 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. PI-lowering-PI.f32 N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f32 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f32 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f32 N/A

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

      13. --lowering--.f32 55.0

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

   8. Simplified 55.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.007000000216066837:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.1809999942779541:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.1809999942779541)
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) zi)
    (*
     (sqrt
      (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
     (fma (* PI (* 2.0 yi)) uy xi)))
   (* xi (cos (* (* uy 2.0) PI)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.1809999942779541f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf((((float) M_PI) * (2.0f * yi)), uy, xi));
	} else {
		tmp = xi * cosf(((uy * 2.0f) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.1809999942779541))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(Float32(Float32(pi) * Float32(Float32(2.0) * yi)), uy, xi)));
	else
		tmp = Float32(xi * cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.18099999

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. *-lowering-*.f32 N/A

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

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.18099999 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.1%

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

   4. Applied egg-rr 97.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot xi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)}, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right)}}} \]

   5. Taylor expanded in xi around inf

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

      1. associate-*l* N/A

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

      2. *-lowering-*.f32 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. cos-lowering-cos.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f32 N/A

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

      7. *-lowering-*.f32 N/A

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

      8. PI-lowering-PI.f32 N/A

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

      9. sqrt-lowering-sqrt.f32 N/A

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f32 N/A

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

   7. Simplified 51.9%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-lowering-*.f32 N/A

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

      2. cos-lowering-cos.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. PI-lowering-PI.f32 51.9

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

   10. Simplified 51.9%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.1809999942779541:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(Float32(Float32(pi) * Float32(Float32(2.0) * yi)), 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. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. *-lowering-*.f32 N/A

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

5. Simplified 80.8%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

6. Final simplification 80.8%

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

Alternative 17: 81.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), Float32(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))) * fma(Float32(Float32(pi) * Float32(Float32(2.0) * yi)), 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. Taylor expanded in uy around 0

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

5. Simplified 80.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)\right)} \]

6. Final simplification 80.8%

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

Alternative 18: 81.7% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Simplified 80.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)\right)} \]

6. Taylor expanded in ux around 0

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

   1. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f32 80.6

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

8. Simplified 80.6%

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

9. Final simplification 80.6%

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

Alternative 19: 81.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, 0 - ux \cdot ux, 1\right)}\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  maxCos
  (* ux (* (- 1.0 ux) zi))
  (*
   (fma (* PI (* 2.0 yi)) uy xi)
   (sqrt (fma (* maxCos maxCos) (- 0.0 (* ux ux)) 1.0)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * ((1.0f - ux) * zi)), (fmaf((((float) M_PI) * (2.0f * yi)), uy, xi) * sqrtf(fmaf((maxCos * maxCos), (0.0f - (ux * ux)), 1.0f))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), Float32(fma(Float32(Float32(pi) * Float32(Float32(2.0) * yi)), uy, xi) * sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(0.0) - Float32(ux * ux)), Float32(1.0)))))
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

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

5. Simplified 80.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)\right)} \]

6. Taylor expanded in ux around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \color{blue}{0 - {ux}^{2}}, 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \mathsf{PI}\left(\right), uy, xi\right)\right) \]

   3. --lowering--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \color{blue}{0 - {ux}^{2}}, 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \mathsf{PI}\left(\right), uy, xi\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, 0 - \color{blue}{ux \cdot ux}, 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \mathsf{PI}\left(\right), uy, xi\right)\right) \]

   5. *-lowering-*.f32 80.5

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, 0 - \color{blue}{ux \cdot ux}, 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)\right) \]

8. Simplified 80.5%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \color{blue}{0 - ux \cdot ux}, 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)\right) \]

9. Final simplification 80.5%

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(\pi \cdot \left(2 \cdot yi\right), uy, xi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, 0 - ux \cdot ux, 1\right)}\right) \]
10. Add Preprocessing

Alternative 20: 81.6% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

5. Simplified 80.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)\right)} \]

6. Taylor expanded in maxCos around 0

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

   1. +-lowering-+.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f32 N/A

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

   8. --lowering--.f32 N/A

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

   9. associate-*r* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. *-lowering-*.f32 N/A

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

   13. PI-lowering-PI.f32 80.4

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

8. Simplified 80.4%

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

9. Final simplification 80.4%

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

Alternative 21: 79.3% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(fma(Float32(uy * Float32(2.0)), Float32(Float32(pi) * yi), xi) + Float32(zi * Float32(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 uy around 0

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

5. Simplified 80.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)\right)} \]

6. Taylor expanded in ux around 0

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

   1. associate-+r+ N/A

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

   2. +-lowering-+.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. PI-lowering-PI.f32 N/A

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

   9. associate-*r* N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-lowering-*.f32 77.7

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

8. Simplified 77.7%

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

9. Final simplification 77.7%

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

Alternative 22: 74.8% accurate, 20.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

5. Simplified 80.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(0 - ux \cdot ux\right), 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot yi\right) \cdot \pi, uy, xi\right)\right)} \]

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-lowering-*.f32 N/A

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

   6. PI-lowering-PI.f32 72.4

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

8. Simplified 72.4%

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

9. Final simplification 72.4%

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

Alternative 23: 48.8% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot xi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)}, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right)}}} \]

5. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. sqrt-lowering-sqrt.f32 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. --lowering--.f32 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. +-lowering-+.f32 N/A

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

   16. *-lowering-*.f32 N/A

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

7. Simplified 49.4%

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

8. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. *-lowering-*.f32 47.2

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

10. Simplified 47.2%

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

Alternative 24: 45.0% accurate, 353.0× speedup

Math

\[\begin{array}{l} \\ xi \end{array} \]

FPCore

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

C

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

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 = xi
end function

Julia

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

MATLAB

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)} \cdot xi, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(ux, \left(0 - \left(1 - ux\right)\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right), 1\right)}, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi, \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)\right)}}} \]

5. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. sqrt-lowering-sqrt.f32 N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f32 N/A

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

   8. *-lowering-*.f32 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. --lowering--.f32 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. +-lowering-+.f32 N/A

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

   16. *-lowering-*.f32 N/A

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

7. Simplified 49.4%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{xi} \]
9. Step-by-step derivation

10. Simplified 42.7%

    \[\leadsto \color{blue}{xi} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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)))