UniformSampleCone, y

Percentage Accurate: 57.2% → 98.3%

Time: 16.3s

Alternatives: 14

Speedup: 5.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in ux around 0

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

5. Simplified 98.2%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt (fma maxCos (* ux (fma ux 2.0 -2.0)) (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(fmaf(maxCos, (ux * fmaf(ux, 2.0f, -2.0f)), (ux * (2.0f - ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(fma(maxCos, Float32(ux * fma(ux, Float32(2.0), Float32(-2.0))), Float32(ux * Float32(Float32(2.0) - ux)))))
end

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in ux around 0

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around 0

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

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

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

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

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

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

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

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

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. --lowering--.f32 97.3

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

8. Simplified 97.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \mathsf{fma}\left(ux, 2, -2\right), ux \cdot \left(2 - ux\right)\right)}} \]
9. Add Preprocessing

Alternative 3: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.019999999552965164:\\ \;\;\;\;uy \cdot \mathsf{fma}\left(2, \pi \cdot t\_0, -1.3333333333333333 \cdot \left(t\_0 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma -2.0 maxCos 2.0))))))
   (if (<= (* uy 2.0) 0.019999999552965164)
     (*
      uy
      (fma
       2.0
       (* PI t_0)
       (* -1.3333333333333333 (* t_0 (* (* uy uy) (* PI (* PI PI)))))))
     (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(-2.0f, maxCos, 2.0f))));
	float tmp;
	if ((uy * 2.0f) <= 0.019999999552965164f) {
		tmp = uy * fmaf(2.0f, (((float) M_PI) * t_0), (-1.3333333333333333f * (t_0 * ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(Float32(-2.0), maxCos, Float32(2.0)))))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.019999999552965164))
		tmp = Float32(uy * fma(Float32(2.0), Float32(Float32(pi) * t_0), Float32(Float32(-1.3333333333333333) * Float32(t_0 * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0199999996

   1. Initial program 60.1%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

   5. Simplified 98.6%

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

   6. Taylor expanded in uy around 0

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

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

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

      2. +-commutative N/A

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

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

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

   8. Simplified 98.6%

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

   if 0.0199999996 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.0%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

   5. Simplified 96.8%

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

   6. Taylor expanded in maxCos around 0

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

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

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

      2. mul-1-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \]

      3. unsub-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

      4. --lowering--.f32 93.9

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

   8. Simplified 93.9%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - ux\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.019999999552965164:\\ \;\;\;\;uy \cdot \mathsf{fma}\left(2, \pi \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)}, -1.3333333333333333 \cdot \left(\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.10000000149011612:\\ \;\;\;\;uy \cdot \mathsf{fma}\left(2, \pi \cdot t\_0, -1.3333333333333333 \cdot \left(t\_0 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma -2.0 maxCos 2.0))))))
   (if (<= (* uy 2.0) 0.10000000149011612)
     (*
      uy
      (fma
       2.0
       (* PI t_0)
       (* -1.3333333333333333 (* t_0 (* (* uy uy) (* PI (* PI PI)))))))
     (* (sin (* (* uy 2.0) PI)) (sqrt (* 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(-2.0f, maxCos, 2.0f))));
	float tmp;
	if ((uy * 2.0f) <= 0.10000000149011612f) {
		tmp = uy * fmaf(2.0f, (((float) M_PI) * t_0), (-1.3333333333333333f * (t_0 * ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(Float32(-2.0), maxCos, Float32(2.0)))))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.10000000149011612))
		tmp = Float32(uy * fma(Float32(2.0), Float32(Float32(pi) * t_0), Float32(Float32(-1.3333333333333333) * Float32(t_0 * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.100000001

   1. Initial program 60.4%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

   5. Simplified 98.6%

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

   6. Taylor expanded in uy around 0

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

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

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

      2. +-commutative N/A

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

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

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

   8. Simplified 96.9%

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

   if 0.100000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 49.2%

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

   3. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f32 40.4

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

   5. Simplified 40.4%

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

   6. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot 2}} \]

      2. *-lowering-*.f32 77.4

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot 2}} \]

   8. Simplified 77.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.10000000149011612:\\ \;\;\;\;uy \cdot \mathsf{fma}\left(2, \pi \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)}, -1.3333333333333333 \cdot \left(\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in ux around 0

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

5. Simplified 98.2%

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

6. Taylor expanded in uy around 0

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

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

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

   2. +-commutative N/A

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

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

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

8. Simplified 87.1%

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

9. Final simplification 87.1%

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

Alternative 6: 87.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;uy \cdot \left(\mathsf{fma}\left(2, \pi \cdot ux, \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(ux \cdot -1.3333333333333333\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(ux \cdot \left(1 - maxCos\right), maxCos + -1, 2\right) + maxCos \cdot -2\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 3.999999989900971e-6)
   (*
    uy
    (*
     (fma
      2.0
      (* PI ux)
      (* (* (* uy uy) (* PI (* PI PI))) (* ux -1.3333333333333333)))
     (sqrt (+ -1.0 (/ 2.0 ux)))))
   (*
    (sqrt
     (*
      ux
      (+ (fma (* ux (- 1.0 maxCos)) (+ maxCos -1.0) 2.0) (* maxCos -2.0))))
    (* 2.0 (* uy PI)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 3.999999989900971e-6f) {
		tmp = uy * (fmaf(2.0f, (((float) M_PI) * ux), (((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * (ux * -1.3333333333333333f))) * sqrtf((-1.0f + (2.0f / ux))));
	} else {
		tmp = sqrtf((ux * (fmaf((ux * (1.0f - maxCos)), (maxCos + -1.0f), 2.0f) + (maxCos * -2.0f)))) * (2.0f * (uy * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(3.999999989900971e-6))
		tmp = Float32(uy * Float32(fma(Float32(2.0), Float32(Float32(pi) * ux), Float32(Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(ux * Float32(-1.3333333333333333)))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(fma(Float32(ux * Float32(Float32(1.0) - maxCos)), Float32(maxCos + Float32(-1.0)), Float32(2.0)) + Float32(maxCos * Float32(-2.0))))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 3.99999999e-6

   1. Initial program 57.3%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

   5. Simplified 98.1%

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

   7. Applied egg-rr 97.8%

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

   8. Taylor expanded in uy around 0

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

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

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

   10. Simplified 86.2%

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

   11. Taylor expanded in maxCos around 0

       \[\leadsto \color{blue}{\left(uy \cdot \left(\frac{-4}{3} \cdot \left(ux \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(ux \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{2 \cdot \frac{1}{ux} - 1}} \]
   12. Step-by-step derivation

       1. associate-*l* N/A

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

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

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

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

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

   13. Simplified 86.0%

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

   if 3.99999999e-6 < maxCos

   1. Initial program 66.5%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

   5. Simplified 60.9%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. neg-mul-1 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. metadata-eval N/A

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

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

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

   7. Applied egg-rr 60.8%

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

   8. Taylor expanded in ux around 0

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

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

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f32 81.8

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

   10. Simplified 81.8%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;uy \cdot \left(\mathsf{fma}\left(2, \pi \cdot ux, \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(ux \cdot -1.3333333333333333\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(ux \cdot \left(1 - maxCos\right), maxCos + -1, 2\right) + maxCos \cdot -2\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (* ux (fma ux (* (+ maxCos -1.0) (- 1.0 maxCos)) (fma maxCos -2.0 2.0))))
  (* uy (fma (* -1.3333333333333333 (* uy uy)) (* PI (* PI PI)) (* 2.0 PI)))))

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in ux around 0

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

5. Simplified 98.2%

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

6. Taylor expanded in uy around 0

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

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 87.1

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

8. Simplified 87.1%

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

9. Final simplification 87.1%

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

Alternative 8: 84.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0010999999940395355:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;uy \cdot \left(\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0010999999940395355)
   (*
    (sqrt
     (* ux (fma ux (* (+ maxCos -1.0) (- 1.0 maxCos)) (fma maxCos -2.0 2.0))))
    (* 2.0 (* uy PI)))
   (*
    uy
    (*
     (sqrt (* ux (fma maxCos -2.0 2.0)))
     (fma -1.3333333333333333 (* (* uy uy) (* PI (* PI PI))) (* 2.0 PI))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0010999999940395355f) {
		tmp = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f)))) * (2.0f * (uy * ((float) M_PI)));
	} else {
		tmp = uy * (sqrtf((ux * fmaf(maxCos, -2.0f, 2.0f))) * fmaf(-1.3333333333333333f, ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0010999999940395355))
		tmp = Float32(sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0))))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(uy * Float32(sqrt(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0)))) * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0011

   1. Initial program 61.3%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

   5. Simplified 98.6%

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

   6. Taylor expanded in uy around 0

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

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

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

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

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

      3. PI-lowering-PI.f32 97.4

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

   8. Simplified 97.4%

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

   if 0.0011 < uy

   1. Initial program 54.2%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-rgt-neg-in N/A

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

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

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

   5. Simplified 97.5%

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

   7. Applied egg-rr 96.8%

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

   8. Taylor expanded in uy around 0

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

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

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

   10. Simplified 68.9%

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

   11. Taylor expanded in ux around 0

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

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

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

       2. metadata-eval N/A

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

       3. cancel-sign-sub-inv N/A

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

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

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

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

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

       6. cancel-sign-sub-inv N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

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

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

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

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

   13. Simplified 59.5%

       \[\leadsto uy \cdot \color{blue}{\left(\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0010999999940395355:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;uy \cdot \left(\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 50.5%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. neg-mul-1 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. metadata-eval N/A

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

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

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

7. Applied egg-rr 49.9%

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

8. Taylor expanded in ux around 0

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

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

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 77.3

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

10. Simplified 77.3%

    \[\leadsto \sqrt{\color{blue}{ux \cdot \left(\mathsf{fma}\left(ux \cdot \left(1 - maxCos\right), maxCos + -1, 2\right) + maxCos \cdot -2\right)}} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
11. Add Preprocessing

Alternative 10: 81.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in ux around 0

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

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

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

   2. cancel-sign-sub-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

   7. distribute-rgt-neg-in N/A

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

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

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

5. Simplified 98.2%

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

6. Taylor expanded in uy around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 77.3

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

8. Simplified 77.3%

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

9. Final simplification 77.3%

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

Alternative 11: 81.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 50.5%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. neg-mul-1 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. metadata-eval N/A

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

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

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

7. Applied egg-rr 49.9%

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

8. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   7. distribute-lft-out N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. +-commutative N/A

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

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

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

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

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

   17. --lowering--.f32 77.2

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

10. Simplified 77.2%

    \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos + -1, 1\right), 1 - maxCos\right)}} \]
11. Add Preprocessing

Alternative 12: 80.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 50.5%

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

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

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

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

   5. associate-*r* N/A

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

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

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

   7. mul-1-neg N/A

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

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

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

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

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

   16. +-commutative N/A

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

   17. *-commutative N/A

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

8. Simplified 48.4%

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

9. Taylor expanded in ux around 0

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

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

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

    2. +-commutative N/A

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

    3. +-commutative N/A

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

    4. associate-+l+ N/A

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

    5. +-commutative N/A

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

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

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

    7. sub-neg N/A

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

    8. metadata-eval N/A

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

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

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

    10. +-commutative N/A

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

    11. accelerator-lowering-fma.f32 76.6

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

11. Simplified 76.6%

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

12. Final simplification 76.6%

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

Alternative 13: 77.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

5. Simplified 50.5%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. neg-mul-1 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. metadata-eval N/A

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

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

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

7. Applied egg-rr 49.9%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux + ux \cdot \left(1 + -1 \cdot ux\right)}\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

   11. --lowering--.f32 73.3

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

10. Simplified 73.3%

    \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 1 - ux, ux\right)}} \]
11. Add Preprocessing

Alternative 14: 63.5% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* 2.0 (* uy PI)) (sqrt (* 2.0 ux))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(2.0) * ux)))
end

MATLAB

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

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in ux around 0

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

   1. cancel-sign-sub-inv N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)}} \]

   2. metadata-eval N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 76.1

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

5. Simplified 76.1%

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

6. Taylor expanded in uy around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 62.4

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

8. Simplified 62.4%

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

9. Taylor expanded in maxCos around 0

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

    1. *-lowering-*.f32 60.4

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

11. Simplified 60.4%

    \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot ux}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))