Trowbridge-Reitz Sample, sample surface normal, cosTheta

Average Error: 0.2 → 0.2

Time: 23.2s

Precision: binary32

Cost: 49824

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

Math

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

↓

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

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 (* PI 2.0)) (* PI 0.5))))))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/
           (/
            1.0
            (+
             1.0
             (pow (* (/ alphay alphax) (tan (* PI (fma 2.0 u1 0.5)))) 2.0)))
           (* 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 * (((float) M_PI) * 2.0f)) + (((float) M_PI) * 0.5f))))));
	return 1.0f / sqrtf((1.0f + (((1.0f / (((1.0f / (1.0f + powf(((alphay / alphax) * tanf((((float) M_PI) * fmaf(2.0f, u1, 0.5f)))), 2.0f))) / (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(pi) * Float32(2.0))) + 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(Float32(1.0) / Float32(Float32(1.0) + (Float32(Float32(alphay / alphax) * tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5))))) ^ Float32(2.0)))) / 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. Taylor expanded in alphay around 0 0.2

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

3. Simplified 0.2

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\color{blue}{{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}^{2}}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
   Proof
   (pow.f32 (cos.f32 (atan.f32 (*.f32 (/.f32 alphay alphax) (tan.f32 (*.f32 (PI.f32) (fma.f32 2 u1 1/2)))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (*.f32 (/.f32 alphay alphax) (tan.f32 (*.f32 (PI.f32) (Rewrite<= fma-def_binary32 (+.f32 (*.f32 2 u1) 1/2))))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (*.f32 (/.f32 alphay alphax) (tan.f32 (Rewrite<= distribute-rgt-out_binary32 (+.f32 (*.f32 (*.f32 2 u1) (PI.f32)) (*.f32 1/2 (PI.f32)))))))) 2): 26 points increase in error, 26 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (*.f32 (/.f32 alphay alphax) (tan.f32 (+.f32 (Rewrite<= associate-*r*_binary32 (*.f32 2 (*.f32 u1 (PI.f32)))) (*.f32 1/2 (PI.f32))))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (*.f32 (/.f32 alphay alphax) (tan.f32 (+.f32 (*.f32 2 (Rewrite=> *-commutative_binary32 (*.f32 (PI.f32) u1))) (*.f32 1/2 (PI.f32))))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (Rewrite<= associate-/r/_binary32 (/.f32 alphay (/.f32 alphax (tan.f32 (+.f32 (*.f32 2 (*.f32 (PI.f32) u1)) (*.f32 1/2 (PI.f32))))))))) 2): 1 points increase in error, 0 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 alphay (tan.f32 (+.f32 (*.f32 2 (*.f32 (PI.f32) u1)) (*.f32 1/2 (PI.f32))))) alphax)))) 2): 0 points increase in error, 1 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (/.f32 (*.f32 alphay (tan.f32 (Rewrite=> +-commutative_binary32 (+.f32 (*.f32 1/2 (PI.f32)) (*.f32 2 (*.f32 (PI.f32) u1)))))) alphax))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (/.f32 (*.f32 alphay (tan.f32 (+.f32 (*.f32 1/2 (PI.f32)) (*.f32 2 (Rewrite<= *-commutative_binary32 (*.f32 u1 (PI.f32))))))) alphax))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f32 (cos.f32 (atan.f32 (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 (tan.f32 (+.f32 (*.f32 1/2 (PI.f32)) (*.f32 2 (*.f32 u1 (PI.f32))))) alphay)) alphax))) 2): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.8
Cost	46560

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

Alternative 2
Error	2.4
Cost	43424

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

Reproduce

herbie shell --seed 2022306 
(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))))))