UniformSampleCone, y

Average Accuracy: 57.7% → 98.3%

Time: 18.6s

Precision: binary32

Cost: 23168

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

Math

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

↓

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

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
 (*
  (sin (cbrt (* (pow (* 2.0 PI) 3.0) (pow uy 3.0))))
  (sqrt
   (fma
    ux
    (- (- 2.0 maxCos) maxCos)
    (* (+ maxCos -1.0) (* ux (* ux (- 1.0 maxCos))))))))

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

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)
	return Float32(sin(cbrt(Float32((Float32(Float32(2.0) * Float32(pi)) ^ Float32(3.0)) * (uy ^ Float32(3.0))))) * sqrt(fma(ux, Float32(Float32(Float32(2.0) - maxCos) - maxCos), Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * Float32(ux * Float32(Float32(1.0) - maxCos)))))))
end

Derivation

1. Initial program 57.7%

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

2. Simplified 57.6%

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

   [Start] 57.7	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   associate-*l* [=>] 57.7	\[ \sin \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   cancel-sign-sub-inv [=>] 57.7	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{1 + \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]
   +-commutative [=>] 57.7	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + 1}} \]
   *-commutative [=>] 57.7	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]
   fma-def [=>] 57.8	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 - ux\right) + ux \cdot maxCos, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)}} \]
   +-commutative [=>] 57.8	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{ux \cdot maxCos + \left(1 - ux\right)}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   associate-+r- [=>] 57.8	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\left(ux \cdot maxCos + 1\right) - ux}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   fma-def [=>] 57.8	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)} - ux, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   neg-sub0 [=>] 57.8	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]
   +-commutative [=>] 57.8	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]
   associate-+r- [=>] 57.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]
   associate--r- [=>] 57.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]
   +-commutative [=>] 57.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux + \left(0 - \left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub0-neg [=>] 57.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux + \color{blue}{\left(-\left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub-neg [<=] 57.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux - \left(ux \cdot maxCos + 1\right)}, 1\right)} \]
   fma-def [=>] 57.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)}, 1\right)} \]

3. Taylor expanded in ux around 0 98.3%

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

4. Simplified 98.3%

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

   [Start] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right) + ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)} \]
   +-commutative [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right) + \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)}} \]
   fma-def [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux, \left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)}} \]
   mul-1-neg [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 + \color{blue}{\left(-\left(maxCos - 1\right)\right)}\right) - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   unsub-neg [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{\left(1 - \left(maxCos - 1\right)\right)} - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   associate-+l- [<=] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{\left(\left(1 - maxCos\right) + 1\right)} - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   +-commutative [<=] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{\left(1 + \left(1 - maxCos\right)\right)} - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   associate-+r- [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{\left(\left(1 + 1\right) - maxCos\right)} - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   metadata-eval [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(\color{blue}{2} - maxCos\right) - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   sub-neg [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \color{blue}{\left(maxCos + \left(-1\right)\right)} \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   metadata-eval [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(maxCos + \color{blue}{-1}\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   +-commutative [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \color{blue}{\left(-1 + maxCos\right)} \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   *-commutative [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(-1 + maxCos\right) \cdot \color{blue}{\left({ux}^{2} \cdot \left(1 - maxCos\right)\right)}\right)} \]
   unpow2 [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(-1 + maxCos\right) \cdot \left(\color{blue}{\left(ux \cdot ux\right)} \cdot \left(1 - maxCos\right)\right)\right)} \]
   associate-*l* [=>] 98.3	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(-1 + maxCos\right) \cdot \color{blue}{\left(ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)} \]

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternatives

Alternative 1
Accuracy	98.3%
Cost	13472

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

Alternative 2
Accuracy	98.3%
Cost	13472

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

Alternative 3
Accuracy	98.3%
Cost	10304

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

Alternative 4
Accuracy	96.8%
Cost	10176

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

Alternative 5
Accuracy	95.8%
Cost	10116

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0004199999966658652:\\ \;\;\;\;uy \cdot \left(2 \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 6
Accuracy	95.8%
Cost	10052

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0004199999966658652:\\ \;\;\;\;uy \cdot \left(2 \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

Alternative 7
Accuracy	89.6%
Cost	9988

\[\begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.006000000052154064:\\ \;\;\;\;uy \cdot \left(2 \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) + \left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 8
Accuracy	81.4%
Cost	7104

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

Alternative 9
Accuracy	80.9%
Cost	7040

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

Alternative 10
Accuracy	80.4%
Cost	6848

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

Alternative 11
Accuracy	77.0%
Cost	6720

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

Alternative 12
Accuracy	77.0%
Cost	6720

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

Alternative 13
Accuracy	62.9%
Cost	6656

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

Reproduce

herbie shell --seed 2023122 
(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)))))))