Trowbridge-Reitz Sample, sample surface normal, cosTheta

Average Accuracy: 99.3% → 99.3%

Time: 26.7s

Precision: binary32

Cost: 59488

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

Math

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

↓

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(2 \cdot u1 + 0.5\right)\right)\right)\\ t_1 := \frac{\cos t_0}{alphax}\\ t_2 := \sin t_0\\ \frac{1}{\sqrt{1 + \frac{u0}{\mathsf{fma}\left(t_1, t_1, t_2 \cdot \frac{t_2}{alphay \cdot alphay}\right) \cdot \left(1 - u0\right)}}} \end{array} \]

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 (atan (* (/ alphay alphax) (tan (* PI (+ (* 2.0 u1) 0.5))))))
        (t_1 (/ (cos t_0) alphax))
        (t_2 (sin t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       u0
       (* (fma t_1 t_1 (* t_2 (/ t_2 (* alphay alphay)))) (- 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 = atanf(((alphay / alphax) * tanf((((float) M_PI) * ((2.0f * u1) + 0.5f)))));
	float t_1 = cosf(t_0) / alphax;
	float t_2 = sinf(t_0);
	return 1.0f / sqrtf((1.0f + (u0 / (fmaf(t_1, t_1, (t_2 * (t_2 / (alphay * alphay)))) * (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 = atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(pi) * Float32(Float32(Float32(2.0) * u1) + Float32(0.5))))))
	t_1 = Float32(cos(t_0) / alphax)
	t_2 = sin(t_0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(u0 / Float32(fma(t_1, t_1, Float32(t_2 * Float32(t_2 / Float32(alphay * alphay)))) * Float32(Float32(1.0) - u0))))))
end

Derivation

1. Initial program 99.3%

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

2. Simplified 99.3%

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

   [Start] 99.3

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

3. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	98.3%
Cost	59360

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(2 \cdot u1 + 0.5\right)\right)\right)\\ t_1 := \frac{\cos t_0}{alphax}\\ \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(t_1, t_1, \sin t_0 \cdot \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot 0.5\right)\right)}{alphay \cdot alphay}\right)}}} \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	56256

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(2 \cdot u1 + 0.5\right)\right)\right)\\ t_1 := \sin t_0\\ \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(\frac{\cos t_0}{alphax}, \frac{\frac{1}{\mathsf{hypot}\left(1, \frac{alphay}{alphax} \cdot \tan \left(\pi \cdot 0.5\right)\right)}}{alphax}, t_1 \cdot \frac{t_1}{alphay \cdot alphay}\right)}}} \end{array} \]

Alternative 3
Accuracy	97.7%
Cost	52992

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

Alternative 4
Accuracy	92.6%
Cost	52960

\[\begin{array}{l} t_0 := \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(2 \cdot u1 + 0.5\right)\right)\right)\\ t_1 := \frac{\cos \tan^{-1} \left(\frac{alphay}{\frac{\frac{\frac{alphax}{u1}}{\pi}}{2}}\right)}{alphax}\\ \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(t_1, t_1, t_0 \cdot \frac{t_0}{alphay \cdot alphay}\right)}}} \end{array} \]

Reproduce

herbie shell --seed 2023151 
(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))))))