Trowbridge-Reitz Sample, sample surface normal, cosTheta

Average Error: 0.2 → 0.2

Time: 31.6s

Precision: binary32

Cost: 42912

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

Math

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

↓

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

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

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

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

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / (((cos(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))))) * cos(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi)))))))) / (alphax * alphax)) + ((sin(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))))) * sin(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi)))))))) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

↓

function tmp = code(u0, u1, alphax, alphay)
	t_0 = atan((tan((single(pi) * (single(0.5) - (u1 * single(-2.0))))) * (alphay / alphax)));
	tmp = single(1.0) / sqrt((single(1.0) + (u0 * ((single(1.0) / (single(1.0) - u0)) / (((cos(t_0) ^ single(2.0)) / (alphax ^ single(2.0))) + ((sin(t_0) ^ single(2.0)) / (alphay ^ single(2.0))))))));
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. Simplified 0.2

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

   [Start] 0.2

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

3. Taylor expanded in u1 around -inf 0.2

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

4. Simplified 0.2

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

   [Start] 0.2	\[ \frac{1}{\sqrt{1 + u0 \cdot \frac{1}{\left(\frac{{\cos \tan^{-1} \left(\frac{\tan \left(\left(0.5 - -2 \cdot u1\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphax}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{\tan \left(\left(0.5 - -2 \cdot u1\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphay}^{2}}\right) \cdot \left(1 - u0\right)}}} \]
   rational.json-simplify-46 [=>] 0.2	\[ \frac{1}{\sqrt{1 + u0 \cdot \color{blue}{\frac{\frac{1}{\frac{{\cos \tan^{-1} \left(\frac{\tan \left(\left(0.5 - -2 \cdot u1\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphax}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{\tan \left(\left(0.5 - -2 \cdot u1\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphay}^{2}}}}{1 - u0}}}} \]
   rational.json-simplify-44 [=>] 0.2	\[ \frac{1}{\sqrt{1 + u0 \cdot \color{blue}{\frac{\frac{1}{1 - u0}}{\frac{{\cos \tan^{-1} \left(\frac{\tan \left(\left(0.5 - -2 \cdot u1\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphax}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{\tan \left(\left(0.5 - -2 \cdot u1\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphay}^{2}}}}}} \]

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.7
Cost	23168

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

Alternative 2
Error	0.8
Cost	23040

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

Alternative 3
Error	1.3
Cost	23008

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

Alternative 4
Error	1.6
Cost	19712

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

Alternative 5
Error	2.7
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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))))))