Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5

Average Error: 12.9 → 0.7

Time: 26.1s

Precision: binary32

Cost: 37124

\[\left(\left(\left(\left(0.0001 \leq alphax \land alphax \leq 1\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\right) \land \left(0 \leq cos2phi \land cos2phi \leq 1\right)\right) \land 0 \leq sin2phi\]

Math

\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

↓

\[\begin{array}{l} t_0 := \frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}\\ \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{u0}^{2}}{t_0} + \left(\left(\frac{u0}{t_0} + 0.3333333333333333 \cdot \frac{{u0}^{3}}{t_0}\right) + 0.25 \cdot \frac{{u0}^{4}}{t_0}\right)\\ \end{array} \]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

↓

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (let* ((t_0 (+ (/ sin2phi (pow alphay 2.0)) (/ cos2phi (pow alphax 2.0)))))
   (if (<= (- 1.0 u0) 0.9599999785423279)
     (/
      (- (log (- 1.0 u0)))
      (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay))))
     (+
      (* 0.5 (/ (pow u0 2.0) t_0))
      (+
       (+ (/ u0 t_0) (* 0.3333333333333333 (/ (pow u0 3.0) t_0)))
       (* 0.25 (/ (pow u0 4.0) t_0)))))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

↓

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float t_0 = (sin2phi / powf(alphay, 2.0f)) + (cos2phi / powf(alphax, 2.0f));
	float tmp;
	if ((1.0f - u0) <= 0.9599999785423279f) {
		tmp = -logf((1.0f - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	} else {
		tmp = (0.5f * (powf(u0, 2.0f) / t_0)) + (((u0 / t_0) + (0.3333333333333333f * (powf(u0, 3.0f) / t_0))) + (0.25f * (powf(u0, 4.0f) / t_0)));
	}
	return tmp;
}

Fortran

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    code = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
end function

↓

real(4) function code(alphax, alphay, u0, cos2phi, sin2phi)
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    real(4), intent (in) :: u0
    real(4), intent (in) :: cos2phi
    real(4), intent (in) :: sin2phi
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (sin2phi / (alphay ** 2.0e0)) + (cos2phi / (alphax ** 2.0e0))
    if ((1.0e0 - u0) <= 0.9599999785423279e0) then
        tmp = -log((1.0e0 - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)))
    else
        tmp = (0.5e0 * ((u0 ** 2.0e0) / t_0)) + (((u0 / t_0) + (0.3333333333333333e0 * ((u0 ** 3.0e0) / t_0))) + (0.25e0 * ((u0 ** 4.0e0) / t_0)))
    end if
    code = tmp
end function

Julia

function code(alphax, alphay, u0, cos2phi, sin2phi)
	return Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))))
end

↓

function code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = Float32(Float32(sin2phi / (alphay ^ Float32(2.0))) + Float32(cos2phi / (alphax ^ Float32(2.0))))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u0) <= Float32(0.9599999785423279))
		tmp = Float32(Float32(-log(Float32(Float32(1.0) - u0))) / Float32(Float32(cos2phi / Float32(alphax * alphax)) + Float32(sin2phi / Float32(alphay * alphay))));
	else
		tmp = Float32(Float32(Float32(0.5) * Float32((u0 ^ Float32(2.0)) / t_0)) + Float32(Float32(Float32(u0 / t_0) + Float32(Float32(0.3333333333333333) * Float32((u0 ^ Float32(3.0)) / t_0))) + Float32(Float32(0.25) * Float32((u0 ^ Float32(4.0)) / t_0))));
	end
	return tmp
end

MATLAB

function tmp = code(alphax, alphay, u0, cos2phi, sin2phi)
	tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
end

↓

function tmp_2 = code(alphax, alphay, u0, cos2phi, sin2phi)
	t_0 = (sin2phi / (alphay ^ single(2.0))) + (cos2phi / (alphax ^ single(2.0)));
	tmp = single(0.0);
	if ((single(1.0) - u0) <= single(0.9599999785423279))
		tmp = -log((single(1.0) - u0)) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
	else
		tmp = (single(0.5) * ((u0 ^ single(2.0)) / t_0)) + (((u0 / t_0) + (single(0.3333333333333333) * ((u0 ^ single(3.0)) / t_0))) + (single(0.25) * ((u0 ^ single(4.0)) / t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 1.7

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   if 0.959999979 < (-.f32 1 u0)

   1. Initial program 14.7

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

   2. Taylor expanded in u0 around 0 0.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{{u0}^{2}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \left(0.25 \cdot \frac{{u0}^{4}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \left(0.3333333333333333 \cdot \frac{{u0}^{3}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right)} \]

   3. Simplified 0.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{{u0}^{2}}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}} + \left(\left(\frac{u0}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}} + 0.3333333333333333 \cdot \frac{{u0}^{3}}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}\right) + 0.25 \cdot \frac{{u0}^{4}}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}\right)} \]
      Proof

      [Start] 0.5	\[ 0.5 \cdot \frac{{u0}^{2}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \left(0.25 \cdot \frac{{u0}^{4}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \left(0.3333333333333333 \cdot \frac{{u0}^{3}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right) \]
      rational.json-simplify-1 [=>] 0.5	\[ 0.5 \cdot \frac{{u0}^{2}}{\color{blue}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}} + \left(0.25 \cdot \frac{{u0}^{4}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \left(0.3333333333333333 \cdot \frac{{u0}^{3}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)\right) \]
      rational.json-simplify-1 [=>] 0.5	\[ 0.5 \cdot \frac{{u0}^{2}}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}} + \color{blue}{\left(\left(0.3333333333333333 \cdot \frac{{u0}^{3}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}} + \frac{u0}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right) + 0.25 \cdot \frac{{u0}^{4}}{\frac{cos2phi}{{alphax}^{2}} + \frac{sin2phi}{{alphay}^{2}}}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{u0}^{2}}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}} + \left(\left(\frac{u0}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}} + 0.3333333333333333 \cdot \frac{{u0}^{3}}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}\right) + 0.25 \cdot \frac{{u0}^{4}}{\frac{sin2phi}{{alphay}^{2}} + \frac{cos2phi}{{alphax}^{2}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	10628

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - \left({u0}^{2} \cdot -0.5 + \left({u0}^{3} \cdot -0.3333333333333333 + {u0}^{4} \cdot -0.25\right)\right)}{t_0}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	10628

\[\begin{array}{l} t_0 := \frac{sin2phi}{alphay \cdot alphay}\\ t_1 := \frac{cos2phi}{alphax \cdot alphax}\\ \mathbf{if}\;1 - u0 \leq 0.9599999785423279:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t_1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(u0 - {u0}^{2} \cdot -0.5\right) - \left({u0}^{3} \cdot -0.3333333333333333 + {u0}^{4} \cdot -0.25\right)}{t_0 + t_1}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	7332

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9850000143051147:\\ \;\;\;\;\frac{-\left(\left(1 + \log \left(1 - u0\right)\right) - 1\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(\left(-u0\right) + \left(-0.5 \cdot {u0}^{2} + -0.3333333333333333 \cdot {u0}^{3}\right)\right)}{t_0}\\ \end{array} \]

Alternative 4
Error	1.2
Cost	3972

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9959999918937683:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(\left(-u0\right) + -0.5 \cdot {u0}^{2}\right)}{t_0}\\ \end{array} \]

Alternative 5
Error	3.2
Cost	3844

\[\begin{array}{l} t_0 := \frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}\\ \mathbf{if}\;1 - u0 \leq 0.9998049736022949:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{u0}{t_0}\\ \end{array} \]

Alternative 6
Error	7.5
Cost	416

\[\frac{u0}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}} \]

Reproduce

herbie shell --seed 2023077 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (and (and (and (and (<= 0.0001 alphax) (<= alphax 1.0)) (and (<= 0.0001 alphay) (<= alphay 1.0))) (and (<= 2.328306437e-10 u0) (<= u0 1.0))) (and (<= 0.0 cos2phi) (<= cos2phi 1.0))) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))