?

Average Error: 14.3 → 0.5
Time: 15.9s
Precision: binary32
Cost: 10660

?

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]
\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\left(-1 \cdot \left(u0 \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\\ \end{array} \]
(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))
↓
(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9599999785423279)
   (* (* (- alpha) alpha) (log (- 1.0 u0)))
   (*
    (- alpha)
    (+
     (+
      (* -1.0 (* u0 alpha))
      (+
       (* -0.3333333333333333 (* (pow u0 3.0) alpha))
       (* -0.5 (* (pow u0 2.0) alpha))))
     (* -0.25 (* (pow u0 4.0) alpha))))))
float code(float alpha, float u0) {
	return (-alpha * alpha) * logf((1.0f - u0));
}
↓
float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9599999785423279f) {
		tmp = (-alpha * alpha) * logf((1.0f - u0));
	} else {
		tmp = -alpha * (((-1.0f * (u0 * alpha)) + ((-0.3333333333333333f * (powf(u0, 3.0f) * alpha)) + (-0.5f * (powf(u0, 2.0f) * alpha)))) + (-0.25f * (powf(u0, 4.0f) * alpha)));
	}
	return tmp;
}

real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    code = (-alpha * alpha) * log((1.0e0 - u0))
end function
↓
real(4) function code(alpha, u0)
    real(4), intent (in) :: alpha
    real(4), intent (in) :: u0
    real(4) :: tmp
    if ((1.0e0 - u0) <= 0.9599999785423279e0) then
        tmp = (-alpha * alpha) * log((1.0e0 - u0))
    else
        tmp = -alpha * ((((-1.0e0) * (u0 * alpha)) + (((-0.3333333333333333e0) * ((u0 ** 3.0e0) * alpha)) + ((-0.5e0) * ((u0 ** 2.0e0) * alpha)))) + ((-0.25e0) * ((u0 ** 4.0e0) * alpha)))
    end if
    code = tmp
end function

function code(alpha, u0)
	return Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)))
end
↓
function code(alpha, u0)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9599999785423279))
		tmp = Float32(Float32(Float32(-alpha) * alpha) * log(Float32(Float32(1.0) - u0)));
	else
		tmp = Float32(Float32(-alpha) * Float32(Float32(Float32(Float32(-1.0) * Float32(u0 * alpha)) + Float32(Float32(Float32(-0.3333333333333333) * Float32((u0 ^ Float32(3.0)) * alpha)) + Float32(Float32(-0.5) * Float32((u0 ^ Float32(2.0)) * alpha)))) + Float32(Float32(-0.25) * Float32((u0 ^ Float32(4.0)) * alpha))));
	end
	return tmp
end

function tmp = code(alpha, u0)
	tmp = (-alpha * alpha) * log((single(1.0) - u0));
end
↓
function tmp_2 = code(alpha, u0)
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9599999785423279))
		tmp = (-alpha * alpha) * log((single(1.0) - u0));
	else
		tmp = -alpha * (((single(-1.0) * (u0 * alpha)) + ((single(-0.3333333333333333) * ((u0 ^ single(3.0)) * alpha)) + (single(-0.5) * ((u0 ^ single(2.0)) * alpha)))) + (single(-0.25) * ((u0 ^ single(4.0)) * alpha)));
	end
	tmp_2 = tmp;
end

\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)
↓
\begin{array}{l}
\mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\
\;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\

\mathbf{else}:\\
\;\;\;\;\left(-\alpha\right) \cdot \left(\left(-1 \cdot \left(u0 \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (-.f32 1 u0) < 0.959999979

    1. Initial program 1.0

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

    if 0.959999979 < (-.f32 1 u0)

    1. Initial program 16.6

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

    2. Simplified16.6

      \[\leadsto \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]
      Proof

      [Start]16.6

      \[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

      rational_best_45_simplify-91 [=>]16.6

      \[ \color{blue}{\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)} \]

      rational_best_45_simplify-25 [=>]16.6

      \[ \color{blue}{\left(-\alpha\right) \cdot \left(\log \left(1 - u0\right) \cdot \alpha\right)} \]

      rational_best_45_simplify-91 [=>]16.6

      \[ \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

    3. Taylor expanded in u0 around 0 0.4

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(-0.25 \cdot \left({u0}^{4} \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + \left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right)\right)} \]

    4. Simplified0.4

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\left(-1 \cdot \left(u0 \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)} \]
      Proof

      [Start]0.4

      \[ \left(-\alpha\right) \cdot \left(-0.25 \cdot \left({u0}^{4} \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + \left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right)\right) \]

      rational_best_45_simplify-73 [=>]0.4

      \[ \left(-\alpha\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + \left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)} \]

      rational_best_45_simplify-80 [=>]0.4

      \[ \left(-\alpha\right) \cdot \left(\color{blue}{\left(-1 \cdot \left(u0 \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right)} + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\left(-1 \cdot \left(u0 \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.5
Cost10500
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;{u0}^{2} \cdot \left(\left(\alpha \cdot \alpha\right) \cdot 0.5\right) - \left(\alpha \cdot \alpha\right) \cdot \left(-0.3333333333333333 \cdot {u0}^{3} + \left(-0.25 \cdot {u0}^{4} - u0\right)\right)\\ \end{array} \]
Alternative 2
Error0.5
Cost10372
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 + \left({u0}^{4} \cdot 0.25 + \left({u0}^{3} \cdot 0.3333333333333333 + 0.5 \cdot {u0}^{2}\right)\right)\right)\right)\\ \end{array} \]
Alternative 3
Error0.4
Cost10372
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left(-0.5 \cdot {u0}^{2} + \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right)\right)\\ \end{array} \]
Alternative 4
Error0.6
Cost7108
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right) \cdot \alpha - u0 \cdot \alpha\right)\\ \end{array} \]
Alternative 5
Error0.6
Cost7076
\[\begin{array}{l} t_0 := \left(-\alpha\right) \cdot \alpha\\ \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;t_0 \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\left(-u0\right) + \left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right)\right)\\ \end{array} \]
Alternative 6
Error0.6
Cost7012
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9860000014305115:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 - \left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right)\right)\right)\\ \end{array} \]
Alternative 7
Error1.0
Cost3716
\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.003700000001117587:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot u0\right) - \left(-0.5 \cdot {u0}^{2}\right) \cdot \left(\alpha \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
Alternative 8
Error1.0
Cost3716
\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.003700000001117587:\\ \;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right) - \left(\alpha \cdot \alpha\right) \cdot \left(-0.5 \cdot {u0}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
Alternative 9
Error3.2
Cost3588
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998649954795837:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right)\\ \end{array} \]
Alternative 10
Error1.0
Cost3588
\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.003700000001117587:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 + {u0}^{2} \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
Alternative 11
Error1.0
Cost3588
\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.003700000001117587:\\ \;\;\;\;\left(u0 + {u0}^{2} \cdot 0.5\right) \cdot \left(\alpha \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
Alternative 12
Error3.2
Cost3524
\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.00013499999477062374:\\ \;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \end{array} \]
Alternative 13
Error8.1
Cost160
\[\alpha \cdot \left(u0 \cdot \alpha\right) \]
Alternative 14
Error8.1
Cost160
\[u0 \cdot \left(\alpha \cdot \alpha\right) \]

Error

Reproduce?

herbie shell --seed 2023098 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))