?

Average Error: 42.61% → 1.05%
Time: 20.6s
Precision: binary32
Cost: 16960

?

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]
\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
↓
\[\cos \left(e^{\log \left(uy \cdot \left(2 \cdot \pi\right)\right)}\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(ux \cdot \frac{maxCos + -1}{\frac{1 + maxCos}{1 - maxCos \cdot maxCos}}\right)} \]
(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 (exp (log (* uy (* 2.0 PI)))))
  (sqrt
   (+
    (* ux (* 2.0 (- 1.0 maxCos)))
    (*
     ux
     (*
      ux
      (/ (+ maxCos -1.0) (/ (+ 1.0 maxCos) (- 1.0 (* maxCos 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(expf(logf((uy * (2.0f * ((float) M_PI)))))) * sqrtf(((ux * (2.0f * (1.0f - maxCos))) + (ux * (ux * ((maxCos + -1.0f) / ((1.0f + maxCos) / (1.0f - (maxCos * 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(exp(log(Float32(uy * Float32(Float32(2.0) * Float32(pi)))))) * sqrt(Float32(Float32(ux * Float32(Float32(2.0) * Float32(Float32(1.0) - maxCos))) + Float32(ux * Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) / Float32(Float32(Float32(1.0) + maxCos) / Float32(Float32(1.0) - Float32(maxCos * 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(exp(log((uy * (single(2.0) * single(pi)))))) * sqrt(((ux * (single(2.0) * (single(1.0) - maxCos))) + (ux * (ux * ((maxCos + single(-1.0)) / ((single(1.0) + maxCos) / (single(1.0) - (maxCos * 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(e^{\log \left(uy \cdot \left(2 \cdot \pi\right)\right)}\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(ux \cdot \frac{maxCos + -1}{\frac{1 + maxCos}{1 - maxCos \cdot maxCos}}\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 42.61

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

  2. Simplified42.67

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

    [Start]42.61

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

    associate-*l* [=>]42.61

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

    cancel-sign-sub-inv [=>]42.61

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

    +-commutative [=>]42.61

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

    *-commutative [=>]42.61

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

    fma-def [=>]42.53

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

    +-commutative [=>]42.53

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

    associate-+r- [=>]42.51

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

    fma-def [=>]42.51

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

    neg-sub0 [=>]42.51

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

    +-commutative [=>]42.51

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

    associate-+r- [=>]42.67

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

    associate--r- [=>]42.67

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

    +-commutative [=>]42.67

    \[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux + \left(0 - \left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]

    sub0-neg [=>]42.67

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

    sub-neg [<=]42.67

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

    fma-def [=>]42.67

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

  3. Taylor expanded in ux around -inf 1.02

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

  4. Simplified1.02

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

    [Start]1.02

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

    +-commutative [=>]1.02

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

    mul-1-neg [=>]1.02

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

    unsub-neg [=>]1.02

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

    *-commutative [=>]1.02

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

    mul-1-neg [=>]1.02

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

    sub-neg [<=]1.02

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

    associate-*l* [=>]1.02

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

    unpow2 [=>]1.02

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

    associate-*l* [=>]1.02

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

    mul-1-neg [=>]1.02

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

    sub-neg [<=]1.02

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

  5. Applied egg-rr1.03

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

  6. Applied egg-rr1.05

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

  7. Final simplification1.05

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

Alternatives

Alternative 1
Error10.48%
Cost16420
\[\begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9998350143432617:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos \cdot maxCos + -1\right)\right)}{1 + maxCos}}\\ \end{array} \]
Alternative 2
Error1.05%
Cost13504
\[\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(1 - maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos + -1\right)\right)\right)} \]
Alternative 3
Error1.02%
Cost10432
\[\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(ux \cdot \frac{1 - maxCos}{\frac{-1}{1 - maxCos}}\right)} \]
Alternative 4
Error3.64%
Cost10212
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(1 - maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux - ux \cdot ux} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \]
Alternative 5
Error1.07%
Cost10176
\[\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot \left(ux \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right)} \]
Alternative 6
Error3.66%
Cost10116
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos \cdot maxCos + -1\right)\right)}{1 + maxCos}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux - ux \cdot ux} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \]
Alternative 7
Error3.61%
Cost10052
\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0004299999854993075:\\ \;\;\;\;\sqrt{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos \cdot maxCos + -1\right)\right)}{1 + maxCos}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
Alternative 8
Error20.02%
Cost4000
\[\sqrt{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos \cdot maxCos + -1\right)\right)}{1 + maxCos}} \]
Alternative 9
Error20.06%
Cost3616
\[\sqrt{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)} \]
Alternative 10
Error20.05%
Cost3616
\[\sqrt{\left(1 - maxCos\right) \cdot \left(ux \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right)} \]
Alternative 11
Error26.1%
Cost3556
\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
Alternative 12
Error35.39%
Cost3424
\[\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)} \]
Alternative 13
Error37.8%
Cost3296
\[\sqrt{\frac{ux}{0.5}} \]
Alternative 14
Error93.43%
Cost3232
\[\sqrt{0} \]

Error

Reproduce?

herbie shell --seed 2023089 
(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)))))))