Average Error: 13.6 → 0.5
Time: 10.9s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\begin{array}{l} t_0 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \sqrt[3]{\left(t_0 \cdot {t_0}^{2}\right) \cdot {\left(2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(2, maxCos \cdot ux, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos, maxCos, 1\right)\right)\right)}^{1.5}} \end{array} \]
(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 (sin (* (* uy 2.0) PI))))
   (cbrt
    (*
     (* t_0 (pow t_0 2.0))
     (pow
      (-
       (* 2.0 (fma maxCos (* ux ux) ux))
       (fma 2.0 (* maxCos ux) (* (* ux ux) (fma maxCos maxCos 1.0))))
      1.5)))))
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 = sinf(((uy * 2.0f) * ((float) M_PI)));
	return cbrtf(((t_0 * powf(t_0, 2.0f)) * powf(((2.0f * fmaf(maxCos, (ux * ux), ux)) - fmaf(2.0f, (maxCos * ux), ((ux * ux) * fmaf(maxCos, maxCos, 1.0f)))), 1.5f)));
}

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 = sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	return cbrt(Float32(Float32(t_0 * (t_0 ^ Float32(2.0))) * (Float32(Float32(Float32(2.0) * fma(maxCos, Float32(ux * ux), ux)) - fma(Float32(2.0), Float32(maxCos * ux), Float32(Float32(ux * ux) * fma(maxCos, maxCos, Float32(1.0))))) ^ Float32(1.5))))
end

\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 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\
\sqrt[3]{\left(t_0 \cdot {t_0}^{2}\right) \cdot {\left(2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(2, maxCos \cdot ux, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos, maxCos, 1\right)\right)\right)}^{1.5}}
\end{array}

Error

Bits error versus ux

Bits error versus uy

Bits error versus maxCos

Derivation

  1. Initial program 13.6

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

  2. Taylor expanded in ux around 0 0.5

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

  3. Simplified0.5

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

  4. Applied egg-rr0.5

    \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right)}^{3} \cdot {\left(2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(2, maxCos \cdot ux, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos, maxCos, 1\right)\right)\right)}^{1.5}}} \]

  5. Applied egg-rr0.5

    \[\leadsto \sqrt[3]{\color{blue}{\left({\sin \left(\left(uy \cdot 2\right) \cdot \pi\right)}^{2} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} \cdot {\left(2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(2, maxCos \cdot ux, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos, maxCos, 1\right)\right)\right)}^{1.5}} \]

  6. Final simplification0.5

    \[\leadsto \sqrt[3]{\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot {\sin \left(\left(uy \cdot 2\right) \cdot \pi\right)}^{2}\right) \cdot {\left(2 \cdot \mathsf{fma}\left(maxCos, ux \cdot ux, ux\right) - \mathsf{fma}\left(2, maxCos \cdot ux, \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos, maxCos, 1\right)\right)\right)}^{1.5}} \]

Reproduce

herbie shell --seed 2022153 
(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)))))))