Average Error: 0.2 → 0.2
Time: 12.8s
Precision: binary32
\[\left(\left(\left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 0.5\right)\right) \land \left(0.0001 \leq alphax \land alphax \leq 1\right)\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\]
\[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
↓
\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot 0.5 + u1 \cdot \left(2 \cdot \pi\right)\right)\right)\\ t_1 := \sin t_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{\tan \left(\mathsf{fma}\left(2, \pi \cdot u1, \pi \cdot 0.5\right)\right) \cdot alphay}{alphax}\right) \cdot \cos t_0}{alphax \cdot alphax} + \frac{t_1 \cdot t_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \]
(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (+
    1.0
    (/
     (*
      (/
       1.0
       (+
        (/
         (*
          (cos
           (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
          (cos
           (atan
            (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI)))))))
         (* alphax alphax))
        (/
         (*
          (sin
           (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
          (sin
           (atan
            (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI)))))))
         (* alphay alphay))))
      u0)
     (- 1.0 u0))))))
↓
(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan (* (/ alphay alphax) (tan (+ (* PI 0.5) (* u1 (* 2.0 PI)))))))
        (t_1 (sin t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/
           (*
            (cos
             (atan (/ (* (tan (fma 2.0 (* PI u1) (* PI 0.5))) alphay) alphax)))
            (cos t_0))
           (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))
float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf((1.0f + (((1.0f / (((cosf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))))) * cosf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI)))))))) / (alphax * alphax)) + ((sinf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))))) * sinf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI)))))))) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}
↓
float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf(((alphay / alphax) * tanf(((((float) M_PI) * 0.5f) + (u1 * (2.0f * ((float) M_PI)))))));
	float t_1 = sinf(t_0);
	return 1.0f / sqrtf((1.0f + (((1.0f / (((cosf(atanf(((tanf(fmaf(2.0f, (((float) M_PI) * u1), (((float) M_PI) * 0.5f))) * alphay) / alphax))) * cosf(t_0)) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(cos(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))) * cos(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi)))))))) / Float32(alphax * alphax)) + Float32(Float32(sin(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))) * sin(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi)))))))) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end
↓
function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(pi) * Float32(0.5)) + Float32(u1 * Float32(Float32(2.0) * Float32(pi)))))))
	t_1 = sin(t_0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(cos(atan(Float32(Float32(tan(fma(Float32(2.0), Float32(Float32(pi) * u1), Float32(Float32(pi) * Float32(0.5)))) * alphay) / alphax))) * cos(t_0)) / Float32(alphax * alphax)) + Float32(Float32(t_1 * t_1) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}}
↓
\begin{array}{l}
t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot 0.5 + u1 \cdot \left(2 \cdot \pi\right)\right)\right)\\
t_1 := \sin t_0\\
\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{\tan \left(\mathsf{fma}\left(2, \pi \cdot u1, \pi \cdot 0.5\right)\right) \cdot alphay}{alphax}\right) \cdot \cos t_0}{alphax \cdot alphax} + \frac{t_1 \cdot t_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}}
\end{array}

Error

Bits error versus u0

Bits error versus u1

Bits error versus alphax

Bits error versus alphay

Derivation

  1. Initial program 0.2

    \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

  2. Applied egg-rr0.2

    \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \color{blue}{\left(\frac{\tan \left(\mathsf{fma}\left(2, \pi \cdot u1, \pi \cdot 0.5\right)\right) \cdot alphay}{alphax}\right)} \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

  3. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{\tan \left(\mathsf{fma}\left(2, \pi \cdot u1, \pi \cdot 0.5\right)\right) \cdot alphay}{alphax}\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot 0.5 + u1 \cdot \left(2 \cdot \pi\right)\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot 0.5 + u1 \cdot \left(2 \cdot \pi\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot 0.5 + u1 \cdot \left(2 \cdot \pi\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (u0 u1 alphax alphay)
  :name "Trowbridge-Reitz Sample, sample surface normal, cosTheta"
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 0.5))) (and (<= 0.0001 alphax) (<= alphax 1.0))) (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (/ 1.0 (sqrt (+ 1.0 (/ (* (/ 1.0 (+ (/ (* (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI)))))) (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))) (* alphax alphax)) (/ (* (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI)))))) (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))) (* alphay alphay)))) u0) (- 1.0 u0))))))