Trowbridge-Reitz Sample, sample surface normal, cosTheta

Average Error: 0.2 → 0.2

Time: 23.6s

Precision: binary32

Cost: 75680

\[\left(\left(\left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 0.5\right)\right) \land \left(0.0001 \leq alphax \land alphax \leq 1\right)\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\]

Math

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

↓

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

FPCore

(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
         (sin
          (atan
           (* (/ alphay alphax) (tan (+ (* u1 (* 2.0 PI)) (* PI 0.5))))))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/
           (*
            (log
             (exp
              (cos
               (atan
                (* (/ alphay alphax) (tan (fma 2.0 (* PI u1) (* PI 0.5))))))))
            (/
             1.0
             (hypot 1.0 (* alphay (/ (tan (* PI (fma 2.0 u1 0.5))) alphax)))))
           (* alphax alphax))
          (/ (* t_0 t_0) (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))

C

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 = sinf(atanf(((alphay / alphax) * tanf(((u1 * (2.0f * ((float) M_PI))) + (((float) M_PI) * 0.5f))))));
	return 1.0f / sqrtf((1.0f + (((1.0f / (((logf(expf(cosf(atanf(((alphay / alphax) * tanf(fmaf(2.0f, (((float) M_PI) * u1), (((float) M_PI) * 0.5f)))))))) * (1.0f / hypotf(1.0f, (alphay * (tanf((((float) M_PI) * fmaf(2.0f, u1, 0.5f))) / alphax))))) / (alphax * alphax)) + ((t_0 * t_0) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

Julia

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 = sin(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(u1 * Float32(Float32(2.0) * Float32(pi))) + Float32(Float32(pi) * Float32(0.5)))))))
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(log(exp(cos(atan(Float32(Float32(alphay / alphax) * tan(fma(Float32(2.0), Float32(Float32(pi) * u1), Float32(Float32(pi) * Float32(0.5))))))))) * Float32(Float32(1.0) / hypot(Float32(1.0), Float32(alphay * Float32(tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))) / alphax))))) / Float32(alphax * alphax)) + Float32(Float32(t_0 * t_0) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

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-rr 0.2

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

3. Taylor expanded in alphay around 0 0.2

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

4. Simplified 0.2

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	53056

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

Alternative 2
Error	0.7
Cost	23136

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

Alternative 3
Error	0.7
Cost	23072

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

Alternative 4
Error	0.7
Cost	19872

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

Alternative 5
Error	2.6
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022228 
(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))))))