Average Error: 13.8 → 0.3
Time: 9.1s
Precision: binary32
\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 \cdot ux - ux \cdot \left(ux + 2 \cdot maxCos\right)\right) + \left(maxCos \cdot \left(ux \cdot ux\right)\right) \cdot \left(2 - maxCos\right)} \]
(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt
   (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))
↓
(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt
   (+
    (- (* 2.0 ux) (* ux (+ ux (* 2.0 maxCos))))
    (* (* maxCos (* ux ux)) (- 2.0 maxCos))))))
float code(float ux, float uy, float maxCos) {
	return cosf(((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 cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((((2.0f * ux) - (ux * (ux + (2.0f * maxCos)))) + ((maxCos * (ux * ux)) * (2.0f - maxCos))));
}

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

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

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

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

  2. Taylor expanded in ux around 0 0.3

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

  3. Simplified0.3

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

  4. Applied *-un-lft-identity_binary320.3

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

  5. Applied cancel-sign-sub-inv_binary320.3

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

  6. Applied distribute-lft-in_binary320.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2022131 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))