Trowbridge-Reitz Sample, sample surface normal, cosTheta

Average Accuracy: 99.3% → 99.3%

Time: 1.7min

Precision: binary32

Cost: 42784

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

Math

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

↓

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

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

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

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

Derivation

1. Initial program 99.4%

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

2. Simplified 99.4%

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

   [Start] 99.4

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

3. Taylor expanded in u1 around 0 99.4%

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

4. Simplified 99.4%

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

   [Start] 99.4	\[ \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{{\sin \tan^{-1} \left(\frac{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphay}^{2}} + \frac{{\cos \tan^{-1} \left(\frac{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphax}^{2}}\right)}}} \]
   associate-/r* [=>] 99.4	\[ \frac{1}{\sqrt{1 + \color{blue}{\frac{\frac{u0}{1 - u0}}{\frac{{\sin \tan^{-1} \left(\frac{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphay}^{2}} + \frac{{\cos \tan^{-1} \left(\frac{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right) \cdot alphay}{alphax}\right)}^{2}}{{alphax}^{2}}}}}} \]

5. Applied egg-rr 99.4%

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

   [Start] 99.4	\[ \frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}{alphay}\right)}^{2} + \frac{{\cos \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}^{2}}{alphax \cdot alphax}}}} \]
   unpow2 [=>] 99.4	\[ \frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\color{blue}{\frac{\sin \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}{alphay} \cdot \frac{\sin \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}{alphay}} + \frac{{\cos \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}^{2}}{alphax \cdot alphax}}}} \]
   frac-times [=>] 99.4	\[ \frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\color{blue}{\frac{\sin \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}{alphay \cdot alphay}} + \frac{{\cos \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}^{2}}{alphax \cdot alphax}}}} \]
   pow2 [=>] 99.4	\[ \frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{\color{blue}{{\sin \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}^{2}}}{alphay \cdot alphay} + \frac{{\cos \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}^{2}}{alphax \cdot alphax}}}} \]
   associate-/r/ [=>] 99.4	\[ \frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{{\sin \tan^{-1} \color{blue}{\left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)\right)}}^{2}}{alphay \cdot alphay} + \frac{{\cos \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}^{2}}{alphax \cdot alphax}}}} \]
   *-commutative [=>] 99.4	\[ \frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{{\sin \tan^{-1} \color{blue}{\left(\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right) \cdot \frac{alphay}{alphax}\right)}}^{2}}{alphay \cdot alphay} + \frac{{\cos \tan^{-1} \left(\frac{alphay}{\frac{alphax}{\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}}\right)}^{2}}{alphax \cdot alphax}}}} \]

6. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	36512

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

Alternative 2
Accuracy	99.4%
Cost	36448

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

Alternative 3
Accuracy	98.4%
Cost	33248

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

Alternative 4
Accuracy	97.9%
Cost	29536

\[\frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2}\right)} + -1\right)}}} \]

Alternative 5
Accuracy	97.9%
Cost	23072

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

Alternative 6
Accuracy	97.8%
Cost	19808

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

Alternative 7
Accuracy	91.2%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023157 
(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))))))