Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 100.0%

Time: 43.8s

Alternatives: 1

Speedup: N/A×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/ (* t_2 t_2) (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))));
	float t_1 = sinf(t_0);
	float t_2 = cosf(t_0);
	return 1.0f / sqrtf((1.0f + (((1.0f / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(t_2 * t_2) / Float32(alphax * alphax)) + Float32(Float32(t_1 * t_1) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	t_0 = atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))));
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/ (* t_2 t_2) (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))));
	float t_1 = sinf(t_0);
	float t_2 = cosf(t_0);
	return 1.0f / sqrtf((1.0f + (((1.0f / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(t_2 * t_2) / Float32(alphax * alphax)) + Float32(Float32(t_1 * t_1) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	t_0 = atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))));
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

Alternative 1: 100.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1 \cdot \pi, 2, 0.5 \cdot \pi\right)\right) \cdot \frac{alphay}{alphax}\right)\\ e^{\frac{\mathsf{log1p}\left(\frac{\frac{u0}{\frac{{\sin t\_0}^{2}}{e^{\log alphay \cdot 2}} + \frac{{\cos t\_0}^{2}}{alphax \cdot alphax}}}{1 - u0}\right) \cdot -1}{2}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan (* (tan (fma (* u1 PI) 2.0 (* 0.5 PI))) (/ alphay alphax)))))
   (exp
    (/
     (*
      (log1p
       (/
        (/
         u0
         (+
          (/ (pow (sin t_0) 2.0) (exp (* (log alphay) 2.0)))
          (/ (pow (cos t_0) 2.0) (* alphax alphax))))
        (- 1.0 u0)))
      -1.0)
     2.0))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf((tanf(fmaf((u1 * ((float) M_PI)), 2.0f, (0.5f * ((float) M_PI)))) * (alphay / alphax)));
	return expf(((log1pf(((u0 / ((powf(sinf(t_0), 2.0f) / expf((logf(alphay) * 2.0f))) + (powf(cosf(t_0), 2.0f) / (alphax * alphax)))) / (1.0f - u0))) * -1.0f) / 2.0f));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(tan(fma(Float32(u1 * Float32(pi)), Float32(2.0), Float32(Float32(0.5) * Float32(pi)))) * Float32(alphay / alphax)))
	return exp(Float32(Float32(log1p(Float32(Float32(u0 / Float32(Float32((sin(t_0) ^ Float32(2.0)) / exp(Float32(log(alphay) * Float32(2.0)))) + Float32((cos(t_0) ^ Float32(2.0)) / Float32(alphax * alphax)))) / Float32(Float32(1.0) - u0))) * Float32(-1.0)) / Float32(2.0)))
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. Add Preprocessing

3. Taylor expanded in u1 around 0

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 100.0%

   \[\leadsto e^{\frac{\mathsf{log1p}\left(\frac{\frac{u0}{\frac{{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1 \cdot \pi, 2, 0.5 \cdot \pi\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2}}{alphay \cdot alphay} + \frac{{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1 \cdot \pi, 2, 0.5 \cdot \pi\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2}}{alphax \cdot alphax}}}{1 - u0}\right) \cdot -1}{2}} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. pow2 N/A

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

    3. pow-to-exp N/A

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

    4. lower-exp.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lower-log.f32 100.0

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

11. Applied rewrites 100.0%

    \[\leadsto e^{\frac{\mathsf{log1p}\left(\frac{\frac{u0}{\frac{{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1 \cdot \pi, 2, 0.5 \cdot \pi\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2}}{e^{\log alphay \cdot 2}} + \frac{{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1 \cdot \pi, 2, 0.5 \cdot \pi\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2}}{alphax \cdot alphax}}}{1 - u0}\right) \cdot -1}{2}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025062 -o reduce:regimes -o reduce:simplify
(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))))))