Average Error: 13.8 → 0.5
Time: 22.7s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sin \left(\sqrt[3]{\left(\left(uy \cdot \left(uy \cdot uy\right)\right) \cdot 8\right) \cdot \left(\pi \cdot \left(\sqrt[3]{\pi} \cdot \left(\pi \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right)\right)\right)}\right) \cdot \sqrt{\left(2 \cdot \left(maxCos \cdot {ux}^{2}\right) + 2 \cdot ux\right) - \left({ux}^{2} + \left({ux}^{2} \cdot {maxCos}^{2} + 2 \cdot \left(maxCos \cdot ux\right)\right)\right)} \]
(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
   (cbrt
    (*
     (* (* uy (* uy uy)) 8.0)
     (* PI (* (cbrt PI) (* PI (* (cbrt PI) (cbrt PI))))))))
  (sqrt
   (-
    (+ (* 2.0 (* maxCos (pow ux 2.0))) (* 2.0 ux))
    (+
     (pow ux 2.0)
     (+ (* (pow ux 2.0) (pow maxCos 2.0)) (* 2.0 (* maxCos ux))))))))
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((((uy * (uy * uy)) * 8.0f) * (((float) M_PI) * (cbrtf(((float) M_PI)) * (((float) M_PI) * (cbrtf(((float) M_PI)) * cbrtf(((float) M_PI))))))))) * sqrtf((((2.0f * (maxCos * powf(ux, 2.0f))) + (2.0f * ux)) - (powf(ux, 2.0f) + ((powf(ux, 2.0f) * powf(maxCos, 2.0f)) + (2.0f * (maxCos * ux))))));
}

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(uy * Float32(uy * uy)) * Float32(8.0)) * Float32(Float32(pi) * Float32(cbrt(Float32(pi)) * Float32(Float32(pi) * Float32(cbrt(Float32(pi)) * cbrt(Float32(pi))))))))) * sqrt(Float32(Float32(Float32(Float32(2.0) * Float32(maxCos * (ux ^ Float32(2.0)))) + Float32(Float32(2.0) * ux)) - Float32((ux ^ Float32(2.0)) + Float32(Float32((ux ^ Float32(2.0)) * (maxCos ^ Float32(2.0))) + Float32(Float32(2.0) * Float32(maxCos * ux)))))))
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)}
↓
\sin \left(\sqrt[3]{\left(\left(uy \cdot \left(uy \cdot uy\right)\right) \cdot 8\right) \cdot \left(\pi \cdot \left(\sqrt[3]{\pi} \cdot \left(\pi \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right)\right)\right)}\right) \cdot \sqrt{\left(2 \cdot \left(maxCos \cdot {ux}^{2}\right) + 2 \cdot ux\right) - \left({ux}^{2} + \left({ux}^{2} \cdot {maxCos}^{2} + 2 \cdot \left(maxCos \cdot ux\right)\right)\right)}

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 13.8

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

  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. Applied add-cbrt-cube_binary320.5

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

  4. Applied add-cbrt-cube_binary320.5

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

  5. Applied add-cbrt-cube_binary320.6

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

  6. Applied cbrt-unprod_binary320.6

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

  7. Applied cbrt-unprod_binary320.6

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

  8. Applied add-cube-cbrt_binary320.5

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

  9. Applied associate-*r*_binary320.5

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

  10. Final simplification0.5

    \[\leadsto \sin \left(\sqrt[3]{\left(\left(uy \cdot \left(uy \cdot uy\right)\right) \cdot 8\right) \cdot \left(\pi \cdot \left(\sqrt[3]{\pi} \cdot \left(\pi \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right)\right)\right)\right)}\right) \cdot \sqrt{\left(2 \cdot \left(maxCos \cdot {ux}^{2}\right) + 2 \cdot ux\right) - \left({ux}^{2} + \left({ux}^{2} \cdot {maxCos}^{2} + 2 \cdot \left(maxCos \cdot ux\right)\right)\right)} \]

Reproduce

herbie shell --seed 2022131 
(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)))))))