UniformSampleCone, y

Average Error: 13.7 → 0.6

Time: 29.0s

Precision: binary32

Cost: 17792

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

Math

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

↓

\[\begin{array}{l} t_0 := -1 + \left(maxCos + \left(-\left(maxCos + \left(maxCos \cdot 2 - 3\right)\right)\right)\right)\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{t_0 \cdot ux + \left(-1 \cdot \left(\left({\left(-1 + maxCos\right)}^{2} + \left(-1 + maxCos\right) \cdot \left(maxCos \cdot 2 - 2\right)\right) - t_0 \cdot \left(1 - maxCos\right)\right)\right) \cdot {ux}^{2}} \end{array} \]

FPCore

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

↓

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ -1.0 (+ maxCos (- (+ maxCos (- (* maxCos 2.0) 3.0)))))))
   (*
    (sin (* (* uy 2.0) PI))
    (sqrt
     (+
      (* t_0 ux)
      (*
       (*
        -1.0
        (-
         (+
          (pow (+ -1.0 maxCos) 2.0)
          (* (+ -1.0 maxCos) (- (* maxCos 2.0) 2.0)))
         (* t_0 (- 1.0 maxCos))))
       (pow ux 2.0)))))))

C

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

↓

float code(float ux, float uy, float maxCos) {
	float t_0 = -1.0f + (maxCos + -(maxCos + ((maxCos * 2.0f) - 3.0f)));
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((t_0 * ux) + ((-1.0f * ((powf((-1.0f + maxCos), 2.0f) + ((-1.0f + maxCos) * ((maxCos * 2.0f) - 2.0f))) - (t_0 * (1.0f - maxCos)))) * powf(ux, 2.0f))));
}

Julia

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

↓

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(-1.0) + Float32(maxCos + Float32(-Float32(maxCos + Float32(Float32(maxCos * Float32(2.0)) - Float32(3.0))))))
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(t_0 * ux) + Float32(Float32(Float32(-1.0) * Float32(Float32((Float32(Float32(-1.0) + maxCos) ^ Float32(2.0)) + Float32(Float32(Float32(-1.0) + maxCos) * Float32(Float32(maxCos * Float32(2.0)) - Float32(2.0)))) - Float32(t_0 * Float32(Float32(1.0) - maxCos)))) * (ux ^ Float32(2.0))))))
end

MATLAB

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

↓

function tmp = code(ux, uy, maxCos)
	t_0 = single(-1.0) + (maxCos + -(maxCos + ((maxCos * single(2.0)) - single(3.0))));
	tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt(((t_0 * ux) + ((single(-1.0) * ((((single(-1.0) + maxCos) ^ single(2.0)) + ((single(-1.0) + maxCos) * ((maxCos * single(2.0)) - single(2.0)))) - (t_0 * (single(1.0) - maxCos)))) * (ux ^ single(2.0)))));
end

Derivation

1. Initial program 13.7

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

2. Applied egg-rr 13.7

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

3. Applied egg-rr 14.0

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

4. Taylor expanded in ux around 0 0.6

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

5. Simplified 0.6

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

   [Start] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2} + \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right) \cdot ux} \]
   rational.json-simplify-1 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right) \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}}} \]
   rational.json-simplify-16 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) + -1\right)} \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}} \]
   rational.json-simplify-1 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(-1 + \left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right)\right)} \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}} \]
   rational.json-simplify-1 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(-1 + \color{blue}{\left(maxCos + -1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right)\right)}\right) \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}} \]
   rational.json-simplify-2 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(-1 + \left(maxCos + \color{blue}{\left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) \cdot -1}\right)\right) \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}} \]
   rational.json-simplify-9 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(-1 + \left(maxCos + \color{blue}{\left(-\left(\left(maxCos + 2 \cdot maxCos\right) - 3\right)\right)}\right)\right) \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}} \]
   rational.json-simplify-1 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(-1 + \left(maxCos + \left(-\left(\color{blue}{\left(2 \cdot maxCos + maxCos\right)} - 3\right)\right)\right)\right) \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}} \]
   rational.json-simplify-48 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(-1 + \left(maxCos + \left(-\color{blue}{\left(maxCos + \left(2 \cdot maxCos - 3\right)\right)}\right)\right)\right) \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}} \]
   rational.json-simplify-2 [=>] 0.6	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(-1 + \left(maxCos + \left(-\left(maxCos + \left(\color{blue}{maxCos \cdot 2} - 3\right)\right)\right)\right)\right) \cdot ux + \left(-1 \cdot \left({\left(maxCos - 1\right)}^{2} + \left(maxCos - 1\right) \cdot \left(2 \cdot maxCos - 2\right)\right) - -1 \cdot \left(\left(1 - maxCos\right) \cdot \left(\left(-1 \cdot \left(\left(maxCos + 2 \cdot maxCos\right) - 3\right) + maxCos\right) - 1\right)\right)\right) \cdot {ux}^{2}} \]

6. Final simplification 0.6

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(-1 + \left(maxCos + \left(-\left(maxCos + \left(maxCos \cdot 2 - 3\right)\right)\right)\right)\right) \cdot ux + \left(-1 \cdot \left(\left({\left(-1 + maxCos\right)}^{2} + \left(-1 + maxCos\right) \cdot \left(maxCos \cdot 2 - 2\right)\right) - \left(-1 + \left(maxCos + \left(-\left(maxCos + \left(maxCos \cdot 2 - 3\right)\right)\right)\right)\right) \cdot \left(1 - maxCos\right)\right)\right) \cdot {ux}^{2}} \]

Alternatives

Alternative 1
Error	2.6
Cost	32008

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := 1 + \left(maxCos \cdot ux - ux\right)\\ t_2 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ t_3 := t_2 \cdot \sqrt{1 - t_0 \cdot t_0}\\ t_4 := t_1 \cdot \left(ux - \left(1 + maxCos \cdot ux\right)\right)\\ \mathbf{if}\;t_3 \leq 1.0999999739205735 \cdot 10^{-12}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(-\left(maxCos + \left(maxCos + -2\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sqrt{1 - t_1 \cdot \frac{t_1 \cdot t_4}{t_4}}\\ \end{array} \]

Alternative 2
Error	2.6
Cost	30984

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := 1 + \left(maxCos \cdot ux - ux\right)\\ t_2 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ t_3 := t_2 \cdot \sqrt{1 - t_0 \cdot t_0}\\ \mathbf{if}\;t_3 \leq 1.0999999739205735 \cdot 10^{-12}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(-\left(maxCos + \left(maxCos + -2\right)\right)\right)}\\ \mathbf{elif}\;t_3 \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sqrt{1 - t_1 \cdot t_1}\\ \end{array} \]

Alternative 3
Error	1.4
Cost	26280

\[\begin{array}{l} t_0 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ t_1 := t_0 \cdot \sqrt{ux + \left(ux - {ux}^{2}\right)}\\ \mathbf{if}\;t_0 \leq 1.8000000379103653 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0.002300000051036477:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.6
Cost	13408

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

Alternative 5
Error	3.1
Cost	10340

\[\begin{array}{l} t_0 := \left(2 - 2 \cdot maxCos\right) \cdot ux\\ \mathbf{if}\;uy \cdot 2 \leq 0.009650000371038914:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{t_0 - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{t_0}\\ \end{array} \]

Alternative 6
Error	4.6
Cost	10148

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux + maxCos \cdot \left(ux \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 - {\left(\left(1 - ux\right) + maxCos \cdot ux\right)}^{2}}\right)\\ \end{array} \]

Alternative 7
Error	4.6
Cost	10148

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux + maxCos \cdot \left(ux \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 + \left(maxCos \cdot ux - ux\right)\right)}^{2}} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]

Alternative 8
Error	7.5
Cost	10048

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

Alternative 9
Error	7.5
Cost	10016

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

Alternative 10
Error	7.5
Cost	9984

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

Alternative 11
Error	8.6
Cost	9856

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

Alternative 12
Error	10.8
Cost	6912

\[2 \cdot \frac{uy \cdot \left(\sqrt{ux \cdot \left(2 - \left(maxCos + maxCos\right)\right)} \cdot \left(\pi \cdot 4\right)\right)}{4} \]

Alternative 13
Error	10.8
Cost	6784

\[2 \cdot \left(\pi \cdot \left(uy \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\right)\right) \]

Alternative 14
Error	10.8
Cost	6784

\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - \left(maxCos + maxCos\right)\right)}\right) \]

Alternative 15
Error	11.6
Cost	6656

\[2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux + ux}\right) \]

Alternative 16
Error	29.7
Cost	3616

\[0 \cdot \sqrt{1 - \left(1 + \left(2 \cdot maxCos - 2\right) \cdot ux\right)} \]

Reproduce

herbie shell --seed 2023073 
(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)))))))