Beckmann Distribution sample, tan2theta, alphax == alphay

Average Error: 14.2 → 0.5

Time: 16.7s

Precision: binary32

Cost: 10500

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9596999883651733:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \left(\left(-u0\right) + -0.3333333333333333 \cdot {u0}^{3}\right) + \alpha \cdot \left(-0.5 \cdot {u0}^{2} + -0.25 \cdot {u0}^{4}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9596999883651733)
   (* (- alpha) (* alpha (log (- 1.0 u0))))
   (*
    (- alpha)
    (+
     (* alpha (+ (- u0) (* -0.3333333333333333 (pow u0 3.0))))
     (* alpha (+ (* -0.5 (pow u0 2.0)) (* -0.25 (pow u0 4.0))))))))

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) <= 0.9596999883651733f) {
		tmp = -alpha * (alpha * logf((1.0f - u0)));
	} else {
		tmp = -alpha * ((alpha * (-u0 + (-0.3333333333333333f * powf(u0, 3.0f)))) + (alpha * ((-0.5f * powf(u0, 2.0f)) + (-0.25f * powf(u0, 4.0f)))));
	}
	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) <= 0.9596999883651733e0) then
        tmp = -alpha * (alpha * log((1.0e0 - u0)))
    else
        tmp = -alpha * ((alpha * (-u0 + ((-0.3333333333333333e0) * (u0 ** 3.0e0)))) + (alpha * (((-0.5e0) * (u0 ** 2.0e0)) + ((-0.25e0) * (u0 ** 4.0e0)))))
    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(0.9596999883651733))
		tmp = Float32(Float32(-alpha) * Float32(alpha * log(Float32(Float32(1.0) - u0))));
	else
		tmp = Float32(Float32(-alpha) * Float32(Float32(alpha * Float32(Float32(-u0) + Float32(Float32(-0.3333333333333333) * (u0 ^ Float32(3.0))))) + Float32(alpha * Float32(Float32(Float32(-0.5) * (u0 ^ Float32(2.0))) + Float32(Float32(-0.25) * (u0 ^ Float32(4.0)))))));
	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(0.9596999883651733))
		tmp = -alpha * (alpha * log((single(1.0) - u0)));
	else
		tmp = -alpha * ((alpha * (-u0 + (single(-0.3333333333333333) * (u0 ^ single(3.0))))) + (alpha * ((single(-0.5) * (u0 ^ single(2.0))) + (single(-0.25) * (u0 ^ single(4.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.959699988

   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 0.959699988 < (-.f32 1 u0)

   1. Initial program 16.5

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

   2. Simplified 16.5

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

      [Start] 16.5	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
      rational.json-simplify-2 [=>] 16.5	\[ \color{blue}{\log \left(1 - u0\right) \cdot \left(\left(-\alpha\right) \cdot \alpha\right)} \]
      rational.json-simplify-43 [=>] 16.5	\[ \color{blue}{\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)} \]

   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. Simplified 0.4

      \[\leadsto \left(-\alpha\right) \cdot \color{blue}{\left(\alpha \cdot \left(\left(-u0\right) + -0.3333333333333333 \cdot {u0}^{3}\right) + \alpha \cdot \left(-0.5 \cdot {u0}^{2} + -0.25 \cdot {u0}^{4}\right)\right)} \]
      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.json-simplify-1 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(-0.25 \cdot \left({u0}^{4} \cdot \alpha\right) + \left(-0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right) + \color{blue}{\left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -1 \cdot \left(u0 \cdot \alpha\right)\right)}\right)\right) \]
      rational.json-simplify-41 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(-0.25 \cdot \left({u0}^{4} \cdot \alpha\right) + \color{blue}{\left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + \left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right)\right)\right)}\right) \]
      rational.json-simplify-1 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(-0.25 \cdot \left({u0}^{4} \cdot \alpha\right) + \color{blue}{\left(\left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right)\right) + -0.5 \cdot \left({u0}^{2} \cdot \alpha\right)\right)}\right) \]
      rational.json-simplify-41 [=>] 0.4	\[ \left(-\alpha\right) \cdot \color{blue}{\left(\left(-1 \cdot \left(u0 \cdot \alpha\right) + -0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right)\right) + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right)} \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\left(\color{blue}{u0 \cdot \left(\alpha \cdot -1\right)} + -0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right)\right) + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\left(\color{blue}{\alpha \cdot \left(-1 \cdot u0\right)} + -0.3333333333333333 \cdot \left({u0}^{3} \cdot \alpha\right)\right) + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\left(\alpha \cdot \left(-1 \cdot u0\right) + \color{blue}{{u0}^{3} \cdot \left(\alpha \cdot -0.3333333333333333\right)}\right) + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\left(\alpha \cdot \left(-1 \cdot u0\right) + \color{blue}{\alpha \cdot \left(-0.3333333333333333 \cdot {u0}^{3}\right)}\right) + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-2 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\left(\alpha \cdot \left(-1 \cdot u0\right) + \color{blue}{\left(-0.3333333333333333 \cdot {u0}^{3}\right) \cdot \alpha}\right) + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-51 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\color{blue}{\alpha \cdot \left(-0.3333333333333333 \cdot {u0}^{3} + -1 \cdot u0\right)} + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-1 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\alpha \cdot \color{blue}{\left(-1 \cdot u0 + -0.3333333333333333 \cdot {u0}^{3}\right)} + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-2 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\alpha \cdot \left(\color{blue}{u0 \cdot -1} + -0.3333333333333333 \cdot {u0}^{3}\right) + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-9 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\alpha \cdot \left(\color{blue}{\left(-u0\right)} + -0.3333333333333333 \cdot {u0}^{3}\right) + \left(-0.5 \cdot \left({u0}^{2} \cdot \alpha\right) + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\alpha \cdot \left(\left(-u0\right) + -0.3333333333333333 \cdot {u0}^{3}\right) + \left(\color{blue}{{u0}^{2} \cdot \left(\alpha \cdot -0.5\right)} + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\alpha \cdot \left(\left(-u0\right) + -0.3333333333333333 \cdot {u0}^{3}\right) + \left(\color{blue}{\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right)} + -0.25 \cdot \left({u0}^{4} \cdot \alpha\right)\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(-\alpha\right) \cdot \left(\alpha \cdot \left(\left(-u0\right) + -0.3333333333333333 \cdot {u0}^{3}\right) + \left(\alpha \cdot \left(-0.5 \cdot {u0}^{2}\right) + \color{blue}{{u0}^{4} \cdot \left(\alpha \cdot -0.25\right)}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9596999883651733:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \left(\left(-u0\right) + -0.3333333333333333 \cdot {u0}^{3}\right) + \alpha \cdot \left(-0.5 \cdot {u0}^{2} + -0.25 \cdot {u0}^{4}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	10404

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9596999883651733:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \left(\left(-0.25 \cdot {u0}^{4} + -0.3333333333333333 \cdot {u0}^{3}\right) + \left(-0.5 \cdot {u0}^{2} - u0\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	10372

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9596999883651733:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 - \left(-0.3333333333333333 \cdot {u0}^{3} + \left(-0.25 \cdot {u0}^{4} + -0.5 \cdot {u0}^{2}\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7140

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9879999756813049:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(u0 \cdot \left(-\alpha\right) + \left(-0.3333333333333333 \cdot {u0}^{3} + -0.5 \cdot {u0}^{2}\right) \cdot \alpha\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7044

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9879999756813049:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \left(-0.3333333333333333 \cdot {u0}^{3} + \left(-0.5 \cdot {u0}^{2} - u0\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	7012

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9879999756813049:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right)\right)\\ \end{array} \]

Alternative 6
Error	1.1
Cost	3716

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(u0 \cdot \alpha + {u0}^{2} \cdot \left(\alpha \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7
Error	3.3
Cost	3588

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998700022697449:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u0 \cdot \alpha\right) \cdot \alpha\\ \end{array} \]

Alternative 8
Error	1.1
Cost	3588

\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.004000000189989805:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 + {u0}^{2} \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \end{array} \]

Alternative 9
Error	8.1
Cost	160

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

Alternative 10
Error	8.1
Cost	160

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

Reproduce

herbie shell --seed 2023073 
(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))))