Average Error: 0.2 → 0.2
Time: 29.1s
Precision: binary32
\[\left(\left(\left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 0.5\right)\right) \land \left(0.0001 \leq alphax \land alphax \leq 1\right)\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\]
\[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
↓
\[\begin{array}{l} t_0 := \frac{alphay \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{alphax}\\ t_1 := \frac{1}{\mathsf{hypot}\left(1, t_0\right)}\\ \frac{1}{\sqrt{1 + \frac{\frac{u0}{\mathsf{fma}\left(t_1, \frac{t_1}{alphax \cdot alphax}, \frac{1}{\frac{alphay \cdot alphay}{{\sin \tan^{-1} t_0}^{2}}}\right)}}{1 - u0}}} \end{array} \]
\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 := \frac{alphay \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{alphax}\\
t_1 := \frac{1}{\mathsf{hypot}\left(1, t_0\right)}\\
\frac{1}{\sqrt{1 + \frac{\frac{u0}{\mathsf{fma}\left(t_1, \frac{t_1}{alphax \cdot alphax}, \frac{1}{\frac{alphay \cdot alphay}{{\sin \tan^{-1} t_0}^{2}}}\right)}}{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 (/ (* alphay (tan (* PI (fma 2.0 u1 0.5)))) alphax))
        (t_1 (/ 1.0 (hypot 1.0 t_0))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (/
        u0
        (fma
         t_1
         (/ t_1 (* alphax alphax))
         (/ 1.0 (/ (* alphay alphay) (pow (sin (atan t_0)) 2.0)))))
       (- 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 = (alphay * tanf(((float) M_PI) * fmaf(2.0f, u1, 0.5f))) / alphax;
	float t_1 = 1.0f / hypotf(1.0f, t_0);
	return 1.0f / sqrtf(1.0f + ((u0 / fmaf(t_1, (t_1 / (alphax * alphax)), (1.0f / ((alphay * alphay) / powf(sinf(atanf(t_0)), 2.0f))))) / (1.0f - u0)));
}

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. Simplified0.2

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

  3. Applied cos-atan_binary320.2

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

  4. Simplified0.2

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

  5. Applied cos-atan_binary320.2

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

  6. Simplified0.2

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

  7. Applied clear-num_binary320.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2022068 
(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))))))