?

Average Error: 13.6 → 0.5
Time: 19.0s
Precision: binary32
Cost: 16672

?

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(maxCos + -1\right)}^{2} \cdot {ux}^{2}\right)} \]
(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
   (+
    (* (- 2.0 (* 2.0 maxCos)) ux)
    (- (* (pow (+ maxCos -1.0) 2.0) (pow ux 2.0)))))))
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((((2.0f - (2.0f * maxCos)) * ux) + -(powf((maxCos + -1.0f), 2.0f) * powf(ux, 2.0f))));
}

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(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)) * ux) + Float32(-Float32((Float32(maxCos + Float32(-1.0)) ^ Float32(2.0)) * (ux ^ Float32(2.0)))))))
end

function tmp = code(ux, uy, maxCos)
	tmp = sin(((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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((((single(2.0) - (single(2.0) * maxCos)) * ux) + -(((maxCos + single(-1.0)) ^ single(2.0)) * (ux ^ single(2.0)))));
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(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux + \left(-{\left(maxCos + -1\right)}^{2} \cdot {ux}^{2}\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Simplified0.5

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

    [Start]0.5

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

    rational_best_oopsla_all_46_json_45_simplify-74 [=>]0.5

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

    rational_best_oopsla_all_46_json_45_simplify-92 [=>]0.5

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

    rational_best_oopsla_all_46_json_45_simplify-45 [=>]0.5

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

  4. Final simplification0.5

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

Alternatives

Alternative 1
Error0.5
Cost10304
\[\sqrt{ux - ux \cdot \left(\left(maxCos - 1\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right) + maxCos\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
Alternative 2
Error0.5
Cost10272
\[\sqrt{-\left(maxCos \cdot ux - ux\right) \cdot \left(1 + \left(maxCos \cdot ux + \left(1 - ux\right)\right)\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
Alternative 3
Error0.5
Cost10208
\[\begin{array}{l} t_0 := maxCos \cdot ux - ux\\ \sqrt{-t_0 \cdot \left(t_0 + 2\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \end{array} \]
Alternative 4
Error1.5
Cost10052
\[\begin{array}{l} \mathbf{if}\;maxCos \leq 1.8000000068241206 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux - ux \cdot \left(-1 + ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(\sqrt{ux - ux \cdot \left(maxCos + \left(maxCos + -1\right) \cdot \left(1 + \left(maxCos \cdot ux - ux\right)\right)\right)} \cdot uy\right)\right)\\ \end{array} \]
Alternative 5
Error3.3
Cost9988
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.003599999938160181:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(\sqrt{ux - ux \cdot \left(maxCos + \left(maxCos + -1\right) \cdot \left(1 + \left(maxCos \cdot ux - ux\right)\right)\right)} \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
Alternative 6
Error6.0
Cost7104
\[2 \cdot \left(\pi \cdot \left(\sqrt{ux - ux \cdot \left(maxCos + \left(maxCos + -1\right) \cdot \left(1 + \left(maxCos \cdot ux - ux\right)\right)\right)} \cdot uy\right)\right) \]
Alternative 7
Error10.9
Cost6784
\[uy \cdot \left(\sqrt{2 \cdot \left(ux - maxCos \cdot ux\right)} \cdot \left(2 \cdot \pi\right)\right) \]
Alternative 8
Error11.8
Cost6656
\[\left(uy \cdot \pi\right) \cdot \left(2 \cdot \sqrt{2 \cdot ux}\right) \]

Error

Reproduce?

herbie shell --seed 2023090 
(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)))))))