Beckmann Distribution sample, tan2theta, alphax == alphay

Average Error: 13.9 → 0.5

Time: 10.5s

Precision: binary32

Cost: 16836

\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9700000286102295:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) + {\alpha}^{2} \cdot \left(0.25 \cdot {u0}^{4} + 0.3333333333333333 \cdot {u0}^{3}\right)\\ \end{array} \]

FPCore

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))

↓

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9700000286102295)
   (* (- alpha) (* alpha (log (- 1.0 u0))))
   (+
    (* (pow alpha 2.0) (+ u0 (* 0.5 (pow u0 2.0))))
    (*
     (pow alpha 2.0)
     (+ (* 0.25 (pow u0 4.0)) (* 0.3333333333333333 (pow u0 3.0)))))))

C

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) <= 0.9700000286102295f) {
		tmp = -alpha * (alpha * logf((1.0f - u0)));
	} else {
		tmp = (powf(alpha, 2.0f) * (u0 + (0.5f * powf(u0, 2.0f)))) + (powf(alpha, 2.0f) * ((0.25f * powf(u0, 4.0f)) + (0.3333333333333333f * powf(u0, 3.0f))));
	}
	return tmp;
}

Fortran

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) <= 0.9700000286102295e0) then
        tmp = -alpha * (alpha * log((1.0e0 - u0)))
    else
        tmp = ((alpha ** 2.0e0) * (u0 + (0.5e0 * (u0 ** 2.0e0)))) + ((alpha ** 2.0e0) * ((0.25e0 * (u0 ** 4.0e0)) + (0.3333333333333333e0 * (u0 ** 3.0e0))))
    end if
    code = tmp
end function

Julia

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(0.9700000286102295))
		tmp = Float32(Float32(-alpha) * Float32(alpha * log(Float32(Float32(1.0) - u0))));
	else
		tmp = Float32(Float32((alpha ^ Float32(2.0)) * Float32(u0 + Float32(Float32(0.5) * (u0 ^ Float32(2.0))))) + Float32((alpha ^ Float32(2.0)) * Float32(Float32(Float32(0.25) * (u0 ^ Float32(4.0))) + Float32(Float32(0.3333333333333333) * (u0 ^ Float32(3.0))))));
	end
	return tmp
end

MATLAB

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(0.9700000286102295))
		tmp = -alpha * (alpha * log((single(1.0) - u0)));
	else
		tmp = ((alpha ^ single(2.0)) * (u0 + (single(0.5) * (u0 ^ single(2.0))))) + ((alpha ^ single(2.0)) * ((single(0.25) * (u0 ^ single(4.0))) + (single(0.3333333333333333) * (u0 ^ single(3.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.970000029

   1. Initial program 1.1

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

   2. Simplified 1.1

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

      [Start] 1.1	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
      rational.json-simplify-2 [=>] 1.1	\[ \color{blue}{\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)} \]
      rational.json-simplify-43 [=>] 1.1	\[ \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

   if 0.970000029 < (-.f32 1 u0)

   1. Initial program 16.4

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

   2. Taylor expanded in u0 around 0 0.3

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

   3. Simplified 0.4

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

      [Start] 0.3	\[ 0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + \left(u0 \cdot {\alpha}^{2} + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right)\right) \]
      rational.json-simplify-41 [=>] 0.4	\[ 0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + \color{blue}{\left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + \left(0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right)\right)} \]
      rational.json-simplify-1 [<=] 0.4	\[ 0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + \left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + \color{blue}{\left(u0 \cdot {\alpha}^{2} + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right)}\right) \]
      rational.json-simplify-41 [<=] 0.4	\[ \color{blue}{\left(u0 \cdot {\alpha}^{2} + 0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right)\right) + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right)} \]
      rational.json-simplify-1 [=>] 0.4	\[ \color{blue}{\left(0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right)} + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) \]
      rational.json-simplify-43 [<=] 0.4	\[ \left(\color{blue}{{\alpha}^{2} \cdot \left(0.5 \cdot {u0}^{2}\right)} + u0 \cdot {\alpha}^{2}\right) + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) \]
      rational.json-simplify-51 [=>] 0.4	\[ \color{blue}{{\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right)} + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + 0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right)\right) \]
      rational.json-simplify-1 [=>] 0.4	\[ {\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) + \color{blue}{\left(0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + 0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right)\right)} \]
      rational.json-simplify-2 [=>] 0.4	\[ {\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) + \left(0.3333333333333333 \cdot \color{blue}{\left({\alpha}^{2} \cdot {u0}^{3}\right)} + 0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ {\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) + \left(\color{blue}{{\alpha}^{2} \cdot \left({u0}^{3} \cdot 0.3333333333333333\right)} + 0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right)\right) \]
      rational.json-simplify-2 [=>] 0.4	\[ {\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) + \left({\alpha}^{2} \cdot \left({u0}^{3} \cdot 0.3333333333333333\right) + 0.25 \cdot \color{blue}{\left({\alpha}^{2} \cdot {u0}^{4}\right)}\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ {\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) + \left({\alpha}^{2} \cdot \left({u0}^{3} \cdot 0.3333333333333333\right) + \color{blue}{{\alpha}^{2} \cdot \left({u0}^{4} \cdot 0.25\right)}\right) \]
      rational.json-simplify-2 [=>] 0.4	\[ {\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) + \left({\alpha}^{2} \cdot \left({u0}^{3} \cdot 0.3333333333333333\right) + \color{blue}{\left({u0}^{4} \cdot 0.25\right) \cdot {\alpha}^{2}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9700000286102295:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\alpha}^{2} \cdot \left(u0 + 0.5 \cdot {u0}^{2}\right) + {\alpha}^{2} \cdot \left(0.25 \cdot {u0}^{4} + 0.3333333333333333 \cdot {u0}^{3}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	10436

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9679999947547913:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({u0}^{4} \cdot -0.25 + \left(\left(-u0\right) + \left({u0}^{2} \cdot -0.5 + -0.3333333333333333 \cdot {u0}^{3}\right)\right)\right) \cdot \left(-\alpha \cdot \alpha\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	10372

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9679999947547913:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left({u0}^{2} \cdot -0.5 + \left({u0}^{4} \cdot -0.25 + -0.3333333333333333 \cdot {u0}^{3}\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7012

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 - \left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7012

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right)\right)\\ \end{array} \]

Alternative 5
Error	1.1
Cost	3652

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9980000257492065:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 - -0.5 \cdot {u0}^{2}\right)\right)\\ \end{array} \]

Alternative 6
Error	1.1
Cost	3652

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9980000257492065:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 - -0.5 \cdot {u0}^{2}\right) \cdot \left(\alpha \cdot \alpha\right)\\ \end{array} \]

Alternative 7
Error	3.4
Cost	3588

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998689889907837:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right)\\ \end{array} \]

Alternative 8
Error	8.3
Cost	160

\[\alpha \cdot \left(u0 \cdot \alpha\right) \]

Alternative 9
Error	8.3
Cost	160

\[u0 \cdot \left(\alpha \cdot \alpha\right) \]

Reproduce

herbie shell --seed 2023064 
(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))))