Trowbridge-Reitz Sample, sample surface normal, cosTheta

Average Error: 0.66% → 0.64%

Time: 44.6s

Precision: binary32

Cost: 42688

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\frac{alphax}{alphay}}\\ \frac{1}{\sqrt{1 + \frac{u0}{{\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} t_0}{alphay}, \frac{1}{alphax \cdot \mathsf{hypot}\left(1, t_0\right)}\right)}\right)}^{6} \cdot \left(1 - u0\right)}}} \end{array} \]

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 (* PI (fma 2.0 u1 0.5))) (/ alphax alphay))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       u0
       (*
        (pow
         (cbrt
          (hypot
           (/ (sin (atan t_0)) alphay)
           (/ 1.0 (* alphax (hypot 1.0 t_0)))))
         6.0)
        (- 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 = tanf((((float) M_PI) * fmaf(2.0f, u1, 0.5f))) / (alphax / alphay);
	return 1.0f / sqrtf((1.0f + (u0 / (powf(cbrtf(hypotf((sinf(atanf(t_0)) / alphay), (1.0f / (alphax * hypotf(1.0f, t_0))))), 6.0f) * (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 = Float32(tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))) / Float32(alphax / alphay))
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(u0 / Float32((cbrt(hypot(Float32(sin(atan(t_0)) / alphay), Float32(Float32(1.0) / Float32(alphax * hypot(Float32(1.0), t_0))))) ^ Float32(6.0)) * Float32(Float32(1.0) - u0))))))
end

Derivation

1. Initial program 0.66

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

2. Simplified 0.66

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

   [Start] 0.66

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

3. Applied egg-rr 0.66

   \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphax}\right)\right)}^{2}\right)\right)} \cdot \left(1 - u0\right)}}} \]

4. Applied egg-rr 0.67

   \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\color{blue}{{\left({\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphax}\right)}\right)}^{2}\right)}^{3}} \cdot \left(1 - u0\right)}}} \]

5. Applied egg-rr 0.65

   \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{{\left({\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{\color{blue}{\frac{1}{\mathsf{hypot}\left(1, \frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}}}{alphax}\right)}\right)}^{2}\right)}^{3} \cdot \left(1 - u0\right)}}} \]

6. Simplified 0.65

   \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{{\left({\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{\color{blue}{\frac{1}{\mathsf{hypot}\left(1, \frac{alphay}{alphax} \cdot \tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)\right)}}}{alphax}\right)}\right)}^{2}\right)}^{3} \cdot \left(1 - u0\right)}}} \]
   Proof

   [Start] 0.65

   \[ \frac{1}{\sqrt{1 + \frac{u0}{{\left({\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{\frac{1}{\mathsf{hypot}\left(1, \frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}}{alphax}\right)}\right)}^{2}\right)}^{3} \cdot \left(1 - u0\right)}}} \]

   *-commutative [=>] 0.65

   \[ \frac{1}{\sqrt{1 + \frac{u0}{{\left({\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{\frac{1}{\mathsf{hypot}\left(1, \frac{alphay}{alphax} \cdot \tan \color{blue}{\left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)}\right)}}{alphax}\right)}\right)}^{2}\right)}^{3} \cdot \left(1 - u0\right)}}} \]

7. Applied egg-rr 0.65

   \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\color{blue}{e^{\log \left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{1}{alphax \cdot \mathsf{hypot}\left(1, \frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}\right)}\right) \cdot 6}} \cdot \left(1 - u0\right)}}} \]

8. Simplified 0.64

   \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\frac{alphax}{alphay}}\right)}{alphay}, \frac{1}{alphax \cdot \mathsf{hypot}\left(1, \frac{\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\frac{alphax}{alphay}}\right)}\right)}\right)}^{6}} \cdot \left(1 - u0\right)}}} \]
   Proof

   [Start] 0.65

   \[ \frac{1}{\sqrt{1 + \frac{u0}{e^{\log \left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{1}{alphax \cdot \mathsf{hypot}\left(1, \frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}\right)}\right) \cdot 6} \cdot \left(1 - u0\right)}}} \]

   exp-to-pow [=>] 0.64

   \[ \frac{1}{\sqrt{1 + \frac{u0}{\color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}, \frac{1}{alphax \cdot \mathsf{hypot}\left(1, \frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}\right)}\right)}^{6}} \cdot \left(1 - u0\right)}}} \]

9. Final simplification 0.64

   \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{{\left(\sqrt[3]{\mathsf{hypot}\left(\frac{\sin \tan^{-1} \left(\frac{\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\frac{alphax}{alphay}}\right)}{alphay}, \frac{1}{alphax \cdot \mathsf{hypot}\left(1, \frac{\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\frac{alphax}{alphay}}\right)}\right)}\right)}^{6} \cdot \left(1 - u0\right)}}} \]

Alternatives

Alternative 1
Error	0.64%
Cost	39488

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

Alternative 2
Error	1.65%
Cost	39328

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

Alternative 3
Error	1.63%
Cost	36288

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

Alternative 4
Error	1.63%
Cost	36224

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

Alternative 5
Error	2.22%
Cost	29472

\[\frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\sin \tan^{-1} \left(\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2}\right)\right)}}} \]

Alternative 6
Error	2.22%
Cost	23072

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

Alternative 7
Error	2.32%
Cost	19872

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

Alternative 8
Error	2.32%
Cost	19808

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

Alternative 9
Error	12.07%
Cost	3680

\[\frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \frac{1}{alphax \cdot alphax}}}} \]

Reproduce

herbie shell --seed 2023089 
(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))))))