Average Error: 0.2 → 0.2
Time: 33.6s
Precision: binary32
\[\left(\left(\left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 0.5\right)\right) \land \left(0.0001 \leq alphax \land alphax \leq 1\right)\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\]
\[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
↓
\[\begin{array}{l} t_0 := \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\\ t_1 := \sin \tan^{-1} t_0\\ t_2 := \frac{1}{\mathsf{hypot}\left(1, t_0\right)}\\ \frac{1}{\sqrt{1 + \frac{u0}{\mathsf{fma}\left(t_2, \frac{t_2}{alphax \cdot alphax}, t_1 \cdot \frac{t_1}{alphay \cdot alphay}\right) \cdot \left(1 - u0\right)}}} \end{array} \]
\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 := \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\\
t_1 := \sin \tan^{-1} t_0\\
t_2 := \frac{1}{\mathsf{hypot}\left(1, t_0\right)}\\
\frac{1}{\sqrt{1 + \frac{u0}{\mathsf{fma}\left(t_2, \frac{t_2}{alphax \cdot alphax}, t_1 \cdot \frac{t_1}{alphay \cdot alphay}\right) \cdot \left(1 - u0\right)}}}
\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 (* (tan (* PI (fma 2.0 u1 0.5))) (/ alphay alphax)))
        (t_1 (sin (atan t_0)))
        (t_2 (/ 1.0 (hypot 1.0 t_0))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       u0
       (*
        (fma t_2 (/ t_2 (* alphax alphax)) (* t_1 (/ t_1 (* alphay alphay))))
        (- 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 = tanf(((float) M_PI) * fmaf(2.0f, u1, 0.5f)) * (alphay / alphax);
	float t_1 = sinf(atanf(t_0));
	float t_2 = 1.0f / hypotf(1.0f, t_0);
	return 1.0f / sqrtf(1.0f + (u0 / (fmaf(t_2, (t_2 / (alphax * alphax)), (t_1 * (t_1 / (alphay * alphay)))) * (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

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

    Proof

  3. Applied cos-atan_binary320.2

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

  4. Simplified0.2

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

    Proof

  5. Applied cos-atan_binary320.2

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

  6. Simplified0.2

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

    Proof

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2022004 
(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))))))