Average Error: 13.5 → 0.5
Time: 37.3s
Precision: binary32
\[2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1 \land 2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1 \land 0 \leq maxCos \land maxCos \leq 1\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \log \left(e \cdot \left(\frac{e}{e^{ux}} \cdot e^{ux \cdot maxCos}\right)\right)} \]
\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)}
↓
\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \log \left(e \cdot \left(\frac{e}{e^{ux}} \cdot e^{ux \cdot maxCos}\right)\right)}

(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 (* (* uy 2.0) PI))
  (sqrt
   (*
    (- ux (* ux maxCos))
    (log (* E (* (/ E (exp ux)) (exp (* ux maxCos)))))))))
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((uy * 2.0f) * ((float) M_PI)) * sqrtf((ux - (ux * maxCos)) * logf(((float) M_E) * ((((float) M_E) / expf(ux)) * expf(ux * maxCos))));
}

Error

Bits error versus ux

Bits error versus uy

Bits error versus maxCos

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.5

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

  2. Simplified0.5

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

  4. Applied add-log-exp_binary320.5

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

  5. Applied add-log-exp_binary320.5

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

  6. Applied add-log-exp_binary320.5

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

  7. Applied add-log-exp_binary320.5

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

  8. Applied diff-log_binary320.5

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

  9. Applied sum-log_binary320.5

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

  10. Applied sum-log_binary320.5

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

  11. Final simplification0.5

    \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot \log \left(e \cdot \left(\frac{e}{e^{ux}} \cdot e^{ux \cdot maxCos}\right)\right)} \]

Reproduce

herbie shell --seed 2021204 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (<= 2.328306437e-10 ux 1.0) (<= 2.328306437e-10 uy 1.0) (<= 0.0 maxCos 1.0))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))