Beckmann Distribution sample, tan2theta, alphax == alphay

Average Error: 14.1 → 0.5

Time: 15.2s

Precision: binary32

Cost: 10372

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

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;1 - u0 \leq 0.9649999737739563:\\ \;\;\;\;\left(t_0 \cdot \left(\alpha \cdot \left(\alpha \cdot \frac{-1}{t_0}\right)\right)\right) \cdot \left(-\left(-t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left({u0}^{2} \cdot 0.5 + \left(u0 - \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha u0)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u0))))
   (if (<= (- 1.0 u0) 0.9649999737739563)
     (* (* t_0 (* alpha (* alpha (/ -1.0 t_0)))) (- (- t_0)))
     (*
      (* alpha alpha)
      (+
       (* (pow u0 2.0) 0.5)
       (-
        u0
        (+ (* -0.3333333333333333 (pow u0 3.0)) (* -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 t_0 = logf((1.0f - u0));
	float tmp;
	if ((1.0f - u0) <= 0.9649999737739563f) {
		tmp = (t_0 * (alpha * (alpha * (-1.0f / t_0)))) * -(-t_0);
	} else {
		tmp = (alpha * alpha) * ((powf(u0, 2.0f) * 0.5f) + (u0 - ((-0.3333333333333333f * powf(u0, 3.0f)) + (-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) :: t_0
    real(4) :: tmp
    t_0 = log((1.0e0 - u0))
    if ((1.0e0 - u0) <= 0.9649999737739563e0) then
        tmp = (t_0 * (alpha * (alpha * ((-1.0e0) / t_0)))) * -(-t_0)
    else
        tmp = (alpha * alpha) * (((u0 ** 2.0e0) * 0.5e0) + (u0 - (((-0.3333333333333333e0) * (u0 ** 3.0e0)) + ((-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)
	t_0 = log(Float32(Float32(1.0) - u0))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9649999737739563))
		tmp = Float32(Float32(t_0 * Float32(alpha * Float32(alpha * Float32(Float32(-1.0) / t_0)))) * Float32(-Float32(-t_0)));
	else
		tmp = Float32(Float32(alpha * alpha) * Float32(Float32((u0 ^ Float32(2.0)) * Float32(0.5)) + Float32(u0 - Float32(Float32(Float32(-0.3333333333333333) * (u0 ^ Float32(3.0))) + 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)
	t_0 = log((single(1.0) - u0));
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9649999737739563))
		tmp = (t_0 * (alpha * (alpha * (single(-1.0) / t_0)))) * -(-t_0);
	else
		tmp = (alpha * alpha) * (((u0 ^ single(2.0)) * single(0.5)) + (u0 - ((single(-0.3333333333333333) * (u0 ^ single(3.0))) + (single(-0.25) * (u0 ^ single(4.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.964999974

   1. Initial program 1.1

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

   2. Applied egg-rr 1.1

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

   3. Applied egg-rr 1.1

      \[\leadsto \color{blue}{0 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\log \left(1 - u0\right) \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right) - \left(\left(\alpha \cdot \alpha\right) \cdot \left(\log \left(1 - u0\right) \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right) \cdot \left(-\log \left(1 - u0\right)\right)} \]

   4. Simplified 1.2

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

      [Start] 1.1	\[ 0 \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\log \left(1 - u0\right) \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right) - \left(\left(\alpha \cdot \alpha\right) \cdot \left(\log \left(1 - u0\right) \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right) \cdot \left(-\log \left(1 - u0\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-102 [=>] 1.1	\[ \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot \left(\log \left(1 - u0\right) \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right) \cdot \left(0 - \left(-\log \left(1 - u0\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.2	\[ \color{blue}{\left(\log \left(1 - u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right)} \cdot \left(0 - \left(-\log \left(1 - u0\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-97 [<=] 1.2	\[ \left(\log \left(1 - u0\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right) \cdot \color{blue}{\left(-\left(-\log \left(1 - u0\right)\right)\right)} \]

   5. Applied egg-rr 1.2

      \[\leadsto \left(\log \left(1 - u0\right) \cdot \color{blue}{\left(\alpha \cdot \left(\alpha \cdot \frac{-1}{\log \left(1 - u0\right)}\right) + 0\right)}\right) \cdot \left(-\left(-\log \left(1 - u0\right)\right)\right) \]

   6. Simplified 1.2

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

      [Start] 1.2	\[ \left(\log \left(1 - u0\right) \cdot \left(\alpha \cdot \left(\alpha \cdot \frac{-1}{\log \left(1 - u0\right)}\right) + 0\right)\right) \cdot \left(-\left(-\log \left(1 - u0\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 1.2	\[ \left(\log \left(1 - u0\right) \cdot \color{blue}{\left(\alpha \cdot \left(\alpha \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right)}\right) \cdot \left(-\left(-\log \left(1 - u0\right)\right)\right) \]

   if 0.964999974 < (-.f32 1 u0)

   1. Initial program 16.6

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

   2. Taylor expanded in u0 around 0 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-1 \cdot u0 + \left(-0.5 \cdot {u0}^{2} + \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.4	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(-0.5 \cdot {u0}^{2} + \left(-1 \cdot u0 + \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.4	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-0.5 \cdot {u0}^{2} + \color{blue}{\left(-0.3333333333333333 \cdot {u0}^{3} + \left(-1 \cdot u0 + -0.25 \cdot {u0}^{4}\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-0.5 \cdot {u0}^{2} + \left(-0.3333333333333333 \cdot {u0}^{3} + \left(\color{blue}{u0 \cdot -1} + -0.25 \cdot {u0}^{4}\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.4	\[ \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(-0.5 \cdot {u0}^{2} + \left(-0.3333333333333333 \cdot {u0}^{3} + \left(\color{blue}{\left(-u0\right)} + -0.25 \cdot {u0}^{4}\right)\right)\right) \]

   4. Applied egg-rr 0.4

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

   5. Simplified 0.4

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

      [Start] 0.4	\[ \left(\alpha \cdot \alpha\right) \cdot \left({u0}^{2} \cdot 0.5\right) + \left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ \color{blue}{\left({u0}^{2} \cdot 0.5\right) \cdot \left(\alpha \cdot \alpha\right)} + \left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.4	\[ \color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left({u0}^{2} \cdot 0.5 + \left(u0 - \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9649999737739563:\\ \;\;\;\;\left(\log \left(1 - u0\right) \cdot \left(\alpha \cdot \left(\alpha \cdot \frac{-1}{\log \left(1 - u0\right)}\right)\right)\right) \cdot \left(-\left(-\log \left(1 - u0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left({u0}^{2} \cdot 0.5 + \left(u0 - \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	10372

\[\begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \mathbf{if}\;1 - u0 \leq 0.9649999737739563:\\ \;\;\;\;\left(t_0 \cdot \left(\alpha \cdot \left(\alpha \cdot \frac{-1}{t_0}\right)\right)\right) \cdot \left(-\left(-t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 + \left(0.5 \cdot {u0}^{2} - \left(-0.3333333333333333 \cdot {u0}^{3} + -0.25 \cdot {u0}^{4}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	7012

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9872000217437744:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{\log \left(1 - u0\right)}{\alpha \cdot \left(-\alpha\right)} - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\left(u0 + \left(0.3333333333333333 \cdot {u0}^{3} + 0.5 \cdot {u0}^{2}\right)\right) \cdot \alpha\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7012

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9872000217437744:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{\log \left(1 - u0\right)}{\alpha \cdot \left(-\alpha\right)} - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(u0 - \left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7012

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9872000217437744:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{\log \left(1 - u0\right)}{\alpha \cdot \left(-\alpha\right)} - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(u0 - -0.3333333333333333 \cdot {u0}^{3}\right) - -0.5 \cdot {u0}^{2}\right)\\ \end{array} \]

Alternative 5
Error	1.1
Cost	3844

\[\begin{array}{l} \mathbf{if}\;u0 \leq 0.004000000189989805:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(u0 + {u0}^{2} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \frac{\log \left(1 - u0\right)}{\alpha \cdot \left(-\alpha\right)} - 0\right)\\ \end{array} \]

Alternative 6
Error	1.1
Cost	3652

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

Alternative 7
Error	3.2
Cost	3588

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998499751091003:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\log \left(1 - u0\right) \cdot \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right)\\ \end{array} \]

Alternative 8
Error	3.2
Cost	3588

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9998499751091003:\\ \;\;\;\;\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right)\\ \end{array} \]

Alternative 9
Error	1.1
Cost	3588

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

Alternative 10
Error	8.2
Cost	160

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

Alternative 11
Error	8.2
Cost	160

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

Reproduce

herbie shell --seed 2023090 
(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))))