Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.8%

Time: 10.0s

Alternatives: 9

Speedup: 2.3×

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

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :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)))
  (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

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :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)))
  (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: 99.8% accurate, 2.3× speedup

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

Math

\[\begin{array}{l} t_0 := {\left(\tan \left(\mathsf{fma}\left(6.2831854820251465, u1, 1.5707963705062866\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} - -1\\ {\left(\frac{u0}{\left(\frac{1 - \frac{1}{t\_0}}{alphay \cdot alphay} - \frac{-1}{t\_0 \cdot \left(alphax \cdot alphax\right)}\right) \cdot \left(1 - u0\right)} - -1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :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)))
  (let* ((t_0
        (-
         (pow
          (*
           (tan (fma 6.2831854820251465 u1 1.5707963705062866))
           (/ alphay alphax))
          2.0)
         -1.0)))
  (pow
   (-
    (/
     u0
     (*
      (-
       (/ (- 1.0 (/ 1.0 t_0)) (* alphay alphay))
       (/ -1.0 (* t_0 (* alphax alphax))))
      (- 1.0 u0)))
    -1.0)
   -0.5)))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = powf((tanf(fmaf(6.2831854820251465f, u1, 1.5707963705062866f)) * (alphay / alphax)), 2.0f) - -1.0f;
	return powf(((u0 / ((((1.0f - (1.0f / t_0)) / (alphay * alphay)) - (-1.0f / (t_0 * (alphax * alphax)))) * (1.0f - u0))) - -1.0f), -0.5f);
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = Float32((Float32(tan(fma(Float32(6.2831854820251465), u1, Float32(1.5707963705062866))) * Float32(alphay / alphax)) ^ Float32(2.0)) - Float32(-1.0))
	return Float32(Float32(u0 / Float32(Float32(Float32(Float32(Float32(1.0) - Float32(Float32(1.0) / t_0)) / Float32(alphay * alphay)) - Float32(Float32(-1.0) / Float32(t_0 * Float32(alphax * alphax)))) * Float32(Float32(1.0) - u0))) - Float32(-1.0)) ^ Float32(-0.5)
end

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.8%

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

4. Evaluated real constant 99.8%

   \[\leadsto {\left(\frac{u0}{\left(\frac{1 - \frac{1}{{\left(\tan \left(\mathsf{fma}\left(0.5, \pi, u1 \cdot 6.2831854820251465\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} + 1}}{alphay \cdot alphay} + \frac{\frac{1}{{\left(\tan \left(\mathsf{fma}\left(0.5, \pi, u1 \cdot 6.2831854820251465\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} + 1}}{alphax \cdot alphax}\right) \cdot \left(1 - u0\right)} - -1\right)}^{-0.5} \]
5. Step-by-step derivation

6. Applied rewrites 99.8%

   \[\leadsto {\left(\frac{u0}{\left(\frac{1 - \frac{1}{{\left(\tan \left(\mathsf{fma}\left(6.2831854820251465, u1, 0.5 \cdot \pi\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} - -1}}{alphay \cdot alphay} - \frac{-1}{\left({\left(\tan \left(\mathsf{fma}\left(6.2831854820251465, u1, 0.5 \cdot \pi\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} - -1\right) \cdot \left(alphax \cdot alphax\right)}\right) \cdot \left(1 - u0\right)} - -1\right)}^{-0.5} \]

7. Evaluated real constant 99.8%

   \[\leadsto {\left(\frac{u0}{\left(\frac{1 - \frac{1}{{\left(\tan \left(\mathsf{fma}\left(6.2831854820251465, u1, 1.5707963705062866\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} - -1}}{alphay \cdot alphay} - \frac{-1}{\left({\left(\tan \left(\mathsf{fma}\left(6.2831854820251465, u1, 1.5707963705062866\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} - -1\right) \cdot \left(alphax \cdot alphax\right)}\right) \cdot \left(1 - u0\right)} - -1\right)}^{-0.5} \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 2.3× speedup

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

Math

\[\begin{array}{l} t_0 := \frac{1}{{\left(\tan \left(\mathsf{fma}\left(0.5, \pi, u1 \cdot 6.2831854820251465\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} + 1}\\ \frac{1}{\sqrt{\frac{u0}{\left(\frac{1 - t\_0}{alphay \cdot alphay} + \frac{t\_0}{alphax \cdot alphax}\right) \cdot \left(1 - u0\right)} - -1}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :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)))
  (let* ((t_0
        (/
         1.0
         (+
          (pow
           (*
            (tan (fma 0.5 PI (* u1 6.2831854820251465)))
            (/ alphay alphax))
           2.0)
          1.0))))
  (/
   1.0
   (sqrt
    (-
     (/
      u0
      (*
       (+ (/ (- 1.0 t_0) (* alphay alphay)) (/ t_0 (* alphax alphax)))
       (- 1.0 u0)))
     -1.0)))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = 1.0f / (powf((tanf(fmaf(0.5f, ((float) M_PI), (u1 * 6.2831854820251465f))) * (alphay / alphax)), 2.0f) + 1.0f);
	return 1.0f / sqrtf(((u0 / ((((1.0f - t_0) / (alphay * alphay)) + (t_0 / (alphax * alphax))) * (1.0f - u0))) - -1.0f));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = Float32(Float32(1.0) / Float32((Float32(tan(fma(Float32(0.5), Float32(pi), Float32(u1 * Float32(6.2831854820251465)))) * Float32(alphay / alphax)) ^ Float32(2.0)) + Float32(1.0)))
	return Float32(Float32(1.0) / sqrt(Float32(Float32(u0 / Float32(Float32(Float32(Float32(Float32(1.0) - t_0) / Float32(alphay * alphay)) + Float32(t_0 / Float32(alphax * alphax))) * Float32(Float32(1.0) - u0))) - Float32(-1.0))))
end

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.3%

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

4. Evaluated real constant 99.3%

   \[\leadsto \frac{1}{\sqrt{\frac{u0}{\left(\frac{1 - \frac{1}{{\left(\tan \left(\mathsf{fma}\left(0.5, \pi, u1 \cdot 6.2831854820251465\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} + 1}}{alphay \cdot alphay} + \frac{\frac{1}{{\left(\tan \left(\mathsf{fma}\left(0.5, \pi, u1 \cdot 6.2831854820251465\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} + 1}}{alphax \cdot alphax}\right) \cdot \left(1 - u0\right)} - -1}} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 3.6× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :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)))
  (pow
 (-
  (/
   u0
   (*
    (-
     (/ 1.0 (* alphay alphay))
     (/
      -1.0
      (*
       (-
        (pow
         (* (tan (* PI (- 0.5 (* -2.0 u1)))) (/ alphay alphax))
         2.0)
        -1.0)
       (* alphax alphax))))
    (- 1.0 u0)))
  -1.0)
 -0.5))

C

float code(float u0, float u1, float alphax, float alphay) {
	return powf(((u0 / (((1.0f / (alphay * alphay)) - (-1.0f / ((powf((tanf((((float) M_PI) * (0.5f - (-2.0f * u1)))) * (alphay / alphax)), 2.0f) - -1.0f) * (alphax * alphax)))) * (1.0f - u0))) - -1.0f), -0.5f);
}

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(u0 / Float32(Float32(Float32(Float32(1.0) / Float32(alphay * alphay)) - Float32(Float32(-1.0) / Float32(Float32((Float32(tan(Float32(Float32(pi) * Float32(Float32(0.5) - Float32(Float32(-2.0) * u1)))) * Float32(alphay / alphax)) ^ Float32(2.0)) - Float32(-1.0)) * Float32(alphax * alphax)))) * Float32(Float32(1.0) - u0))) - Float32(-1.0)) ^ Float32(-0.5)
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = ((u0 / (((single(1.0) / (alphay * alphay)) - (single(-1.0) / ((((tan((single(pi) * (single(0.5) - (single(-2.0) * u1)))) * (alphay / alphax)) ^ single(2.0)) - single(-1.0)) * (alphax * alphax)))) * (single(1.0) - u0))) - single(-1.0)) ^ single(-0.5);
end

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in alphax around 0

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

6. Applied rewrites 98.9%

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

8. Applied rewrites 98.9%

   \[\leadsto {\left(\frac{u0}{\left(\frac{1}{alphay \cdot alphay} - \frac{-1}{\left({\left(\tan \left(\pi \cdot \left(0.5 - -2 \cdot u1\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} - -1\right) \cdot \left(alphax \cdot alphax\right)}\right) \cdot \left(1 - u0\right)} - -1\right)}^{-0.5} \]
9. Add Preprocessing

Alternative 4: 98.4% accurate, 4.0× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :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
  (-
   (/
    u0
    (*
     (-
      (/ 1.0 (* alphay alphay))
      (/
       -1.0
       (*
        (-
         (pow
          (* (tan (* PI (- 0.5 (* -2.0 u1)))) (/ alphay alphax))
          2.0)
         -1.0)
        (* alphax alphax))))
     (- 1.0 u0)))
   -1.0))))

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf(((u0 / (((1.0f / (alphay * alphay)) - (-1.0f / ((powf((tanf((((float) M_PI) * (0.5f - (-2.0f * u1)))) * (alphay / alphax)), 2.0f) - -1.0f) * (alphax * alphax)))) * (1.0f - u0))) - -1.0f));
}

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(u0 / Float32(Float32(Float32(Float32(1.0) / Float32(alphay * alphay)) - Float32(Float32(-1.0) / Float32(Float32((Float32(tan(Float32(Float32(pi) * Float32(Float32(0.5) - Float32(Float32(-2.0) * u1)))) * Float32(alphay / alphax)) ^ Float32(2.0)) - Float32(-1.0)) * Float32(alphax * alphax)))) * Float32(Float32(1.0) - u0))) - Float32(-1.0))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) / sqrt(((u0 / (((single(1.0) / (alphay * alphay)) - (single(-1.0) / ((((tan((single(pi) * (single(0.5) - (single(-2.0) * u1)))) * (alphay / alphax)) ^ single(2.0)) - single(-1.0)) * (alphax * alphax)))) * (single(1.0) - u0))) - single(-1.0)));
end

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in alphax around 0

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

6. Applied rewrites 98.4%

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

8. Applied rewrites 98.4%

   \[\leadsto \frac{1}{\sqrt{\frac{u0}{\left(\frac{1}{alphay \cdot alphay} - \frac{-1}{\left({\left(\tan \left(\pi \cdot \left(0.5 - -2 \cdot u1\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} - -1\right) \cdot \left(alphax \cdot alphax\right)}\right) \cdot \left(1 - u0\right)} - -1}} \]
9. Add Preprocessing

Alternative 5: 96.8% accurate, 4.1× speedup

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

Math

\[\mathsf{fma}\left(\frac{\left(alphay \cdot alphay\right) \cdot u0}{{\tanh \sinh^{-1} \left(\tan \left(\mathsf{fma}\left(\pi + \pi, u1, 1.5707963705062866\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} \cdot \left(1 - u0\right)}, -0.5, 1\right) \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :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)))
  (fma
 (/
  (* (* alphay alphay) u0)
  (*
   (pow
    (tanh
     (asinh
      (*
       (tan (fma (+ PI PI) u1 1.5707963705062866))
       (/ alphay alphax))))
    2.0)
   (- 1.0 u0)))
 -0.5
 1.0))

C

float code(float u0, float u1, float alphax, float alphay) {
	return fmaf((((alphay * alphay) * u0) / (powf(tanhf(asinhf((tanf(fmaf((((float) M_PI) + ((float) M_PI)), u1, 1.5707963705062866f)) * (alphay / alphax)))), 2.0f) * (1.0f - u0))), -0.5f, 1.0f);
}

Julia

function code(u0, u1, alphax, alphay)
	return fma(Float32(Float32(Float32(alphay * alphay) * u0) / Float32((tanh(asinh(Float32(tan(fma(Float32(Float32(pi) + Float32(pi)), u1, Float32(1.5707963705062866))) * Float32(alphay / alphax)))) ^ Float32(2.0)) * Float32(Float32(1.0) - u0))), Float32(-0.5), Float32(1.0))
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. Taylor expanded in alphay around 0

   \[\leadsto 1 + \frac{-1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \pi + 2 \cdot \left(u1 \cdot \pi\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \pi + 2 \cdot \left(u1 \cdot \pi\right)\right)}\right)}^{2} \cdot \left(1 - u0\right)} \]
3. Step-by-step derivation

4. Applied rewrites 96.8%

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

6. Applied rewrites 96.8%

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

7. Evaluated real constant 96.8%

   \[\leadsto \mathsf{fma}\left(\frac{\left(alphay \cdot alphay\right) \cdot u0}{{\tanh \sinh^{-1} \left(\tan \left(\mathsf{fma}\left(\pi + \pi, u1, 1.5707963705062866\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2} \cdot \left(1 - u0\right)}, -0.5, 1\right) \]
8. Add Preprocessing

Alternative 6: 96.7% accurate, 4.3× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :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)))
  (fma
 (/
  (* u0 (* alphay alphay))
  (*
   (pow (tanh (asinh (* (tan (* 0.5 PI)) (/ alphay alphax)))) 2.0)
   (- 1.0 u0)))
 -0.5
 1.0))

C

float code(float u0, float u1, float alphax, float alphay) {
	return fmaf(((u0 * (alphay * alphay)) / (powf(tanhf(asinhf((tanf((0.5f * ((float) M_PI))) * (alphay / alphax)))), 2.0f) * (1.0f - u0))), -0.5f, 1.0f);
}

Julia

function code(u0, u1, alphax, alphay)
	return fma(Float32(Float32(u0 * Float32(alphay * alphay)) / Float32((tanh(asinh(Float32(tan(Float32(Float32(0.5) * Float32(pi))) * Float32(alphay / alphax)))) ^ Float32(2.0)) * Float32(Float32(1.0) - u0))), Float32(-0.5), Float32(1.0))
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. Taylor expanded in alphay around 0

   \[\leadsto 1 + \frac{-1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \pi + 2 \cdot \left(u1 \cdot \pi\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \pi + 2 \cdot \left(u1 \cdot \pi\right)\right)}\right)}^{2} \cdot \left(1 - u0\right)} \]
3. Step-by-step derivation

4. Applied rewrites 96.8%

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

5. Taylor expanded in u1 around 0

   \[\leadsto 1 + -0.5 \cdot \frac{{alphay}^{2} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \pi\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \pi\right)}\right)}^{2} \cdot \left(1 - u0\right)} \]
6. Step-by-step derivation

7. Applied rewrites 96.7%

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

9. Applied rewrites 96.7%

   \[\leadsto \mathsf{fma}\left(\frac{u0 \cdot \left(alphay \cdot alphay\right)}{{\tanh \sinh^{-1} \left(\tan \left(0.5 \cdot \pi\right) \cdot \frac{alphay}{alphax}\right)}^{2} \cdot \left(1 - u0\right)}, -0.5, 1\right) \]
10. Add Preprocessing

Alternative 7: 92.6% accurate, 9.6× speedup

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

Math

\[{\left(\frac{u0}{\left(\frac{1}{alphax \cdot alphax} + \frac{1}{alphay \cdot alphay}\right) \cdot \left(1 - u0\right)} - -1\right)}^{-0.5} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :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)))
  (pow
 (-
  (/
   u0
   (*
    (+ (/ 1.0 (* alphax alphax)) (/ 1.0 (* alphay alphay)))
    (- 1.0 u0)))
  -1.0)
 -0.5))

C

float code(float u0, float u1, float alphax, float alphay) {
	return powf(((u0 / (((1.0f / (alphax * alphax)) + (1.0f / (alphay * alphay))) * (1.0f - u0))) - -1.0f), -0.5f);
}

Fortran

real(4) function code(u0, u1, alphax, alphay)
use fmin_fmax_functions
    real(4), intent (in) :: u0
    real(4), intent (in) :: u1
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    code = ((u0 / (((1.0e0 / (alphax * alphax)) + (1.0e0 / (alphay * alphay))) * (1.0e0 - u0))) - (-1.0e0)) ** (-0.5e0)
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(u0 / Float32(Float32(Float32(Float32(1.0) / Float32(alphax * alphax)) + Float32(Float32(1.0) / Float32(alphay * alphay))) * Float32(Float32(1.0) - u0))) - Float32(-1.0)) ^ Float32(-0.5)
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = ((u0 / (((single(1.0) / (alphax * alphax)) + (single(1.0) / (alphay * alphay))) * (single(1.0) - u0))) - single(-1.0)) ^ single(-0.5);
end

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in alphax around inf

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

6. Applied rewrites 93.1%

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

7. Taylor expanded in alphax around 0

   \[\leadsto \frac{1}{\sqrt{\frac{u0}{\left(\frac{1}{{alphay}^{2}} + \frac{1}{{alphax}^{2}}\right) \cdot \left(1 - u0\right)} - -1}} \]
8. Step-by-step derivation

9. Applied rewrites 92.5%

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

11. Applied rewrites 92.6%

    \[\leadsto {\left(\frac{u0}{\left(\frac{1}{alphax \cdot alphax} + \frac{1}{alphay \cdot alphay}\right) \cdot \left(1 - u0\right)} - -1\right)}^{-0.5} \]
12. Add Preprocessing

Alternative 8: 92.5% accurate, 12.3× speedup

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

Math

\[\frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{\frac{1}{alphax \cdot alphax} + \frac{1}{alphay \cdot alphay}} - -1}} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :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
  (-
   (/
    (/ u0 (- 1.0 u0))
    (+ (/ 1.0 (* alphax alphax)) (/ 1.0 (* alphay alphay))))
   -1.0))))

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf((((u0 / (1.0f - u0)) / ((1.0f / (alphax * alphax)) + (1.0f / (alphay * alphay)))) - -1.0f));
}

Fortran

real(4) function code(u0, u1, alphax, alphay)
use fmin_fmax_functions
    real(4), intent (in) :: u0
    real(4), intent (in) :: u1
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    code = 1.0e0 / sqrt((((u0 / (1.0e0 - u0)) / ((1.0e0 / (alphax * alphax)) + (1.0e0 / (alphay * alphay)))) - (-1.0e0)))
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(Float32(u0 / Float32(Float32(1.0) - u0)) / Float32(Float32(Float32(1.0) / Float32(alphax * alphax)) + Float32(Float32(1.0) / Float32(alphay * alphay)))) - Float32(-1.0))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) / sqrt((((u0 / (single(1.0) - u0)) / ((single(1.0) / (alphax * alphax)) + (single(1.0) / (alphay * alphay)))) - single(-1.0)));
end

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in alphax around inf

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

6. Applied rewrites 93.1%

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

7. Taylor expanded in alphax around 0

   \[\leadsto \frac{1}{\sqrt{\frac{u0}{\left(\frac{1}{{alphay}^{2}} + \frac{1}{{alphax}^{2}}\right) \cdot \left(1 - u0\right)} - -1}} \]
8. Step-by-step derivation

9. Applied rewrites 92.5%

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

11. Applied rewrites 92.5%

    \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{\frac{1}{alphax \cdot alphax} + \frac{1}{alphay \cdot alphay}} - -1}} \]
12. Add Preprocessing

Alternative 9: 92.5% accurate, 12.6× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :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
  (-
   (/
    u0
    (*
     (- 1.0 u0)
     (+ (/ 1.0 (* alphax alphax)) (/ 1.0 (* alphay alphay)))))
   -1.0))))

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf(((u0 / ((1.0f - u0) * ((1.0f / (alphax * alphax)) + (1.0f / (alphay * alphay))))) - -1.0f));
}

Fortran

real(4) function code(u0, u1, alphax, alphay)
use fmin_fmax_functions
    real(4), intent (in) :: u0
    real(4), intent (in) :: u1
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    code = 1.0e0 / sqrt(((u0 / ((1.0e0 - u0) * ((1.0e0 / (alphax * alphax)) + (1.0e0 / (alphay * alphay))))) - (-1.0e0)))
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(Float32(1.0) / Float32(alphax * alphax)) + Float32(Float32(1.0) / Float32(alphay * alphay))))) - Float32(-1.0))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) / sqrt(((u0 / ((single(1.0) - u0) * ((single(1.0) / (alphax * alphax)) + (single(1.0) / (alphay * alphay))))) - single(-1.0)));
end

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in alphax around inf

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

6. Applied rewrites 93.1%

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

7. Taylor expanded in alphax around 0

   \[\leadsto \frac{1}{\sqrt{\frac{u0}{\left(\frac{1}{{alphay}^{2}} + \frac{1}{{alphax}^{2}}\right) \cdot \left(1 - u0\right)} - -1}} \]
8. Step-by-step derivation

9. Applied rewrites 92.5%

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

11. Applied rewrites 92.5%

    \[\leadsto \frac{1}{\sqrt{\frac{u0}{\left(1 - u0\right) \cdot \left(\frac{1}{alphax \cdot alphax} + \frac{1}{alphay \cdot alphay}\right)} - -1}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2026035 +o sampling:rival3
(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))))))