Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.3% → 99.9%

Time: 7.7s

Alternatives: 8

Speedup: 1.7×

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: 99.3% 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: 99.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (pow
  (-
   (/
    (/
     u0
     (+
      (pow
       (/
        (sin (atan (* (tan (fma 0.5 PI (* u1 (+ PI PI)))) (/ alphay alphax))))
        alphay)
       2.0)
      (pow
       (/
        1.0
        (*
         (sqrt
          (+
           (pow (* (/ alphay alphax) (tan (fma PI 0.5 (* (* 2.0 PI) u1)))) 2.0)
           1.0))
         alphax))
       2.0)))
    (- 1.0 u0))
   -1.0)
  -0.5))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.9%

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

   1. lift-/.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 99.9

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

7. Applied rewrites 99.9%

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

Alternative 2: 99.4% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (+
    1.0
    (/
     u0
     (*
      (+
       (pow
        (/
         (sin (atan (* (tan (fma 0.5 PI (* u1 (+ PI PI)))) (/ alphay alphax))))
         alphay)
        2.0)
       (pow
        (/
         1.0
         (*
          (sqrt
           (+
            (pow
             (* (/ alphay alphax) (tan (fma PI 0.5 (* (* 2.0 PI) u1))))
             2.0)
            1.0))
          alphax))
        2.0))
      (- 1.0 u0)))))))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 99.4

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

7. Applied rewrites 99.4%

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

Alternative 3: 99.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (pow
  (-
   (/
    (/
     u0
     (+
      (pow
       (/
        (sin (atan (* (tan (fma 0.5 PI (* u1 (+ PI PI)))) (/ alphay alphax))))
        alphay)
       2.0)
      (pow
       (/
        1.0
        (* (* (/ alphay alphax) (tan (fma 0.5 PI (* 2.0 (* u1 PI))))) alphax))
       2.0)))
    (- 1.0 u0))
   -1.0)
  -0.5))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.9%

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

   1. lift-/.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in alphax around 0

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

   1. times-frac N/A

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

   2. lift-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. quot-tan N/A

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

   5. lower-tan.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 99.0

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

10. Applied rewrites 99.0%

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

Alternative 4: 98.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (+
    1.0
    (/
     u0
     (*
      (+
       (pow
        (/
         (sin (atan (* (tan (fma 0.5 PI (* u1 (+ PI PI)))) (/ alphay alphax))))
         alphay)
        2.0)
       (pow
        (/
         1.0
         (* (* (/ alphay alphax) (tan (fma 0.5 PI (* 2.0 (* u1 PI))))) alphax))
        2.0))
      (- 1.0 u0)))))))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in alphax around 0

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

   1. times-frac N/A

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

   2. lift-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. quot-tan N/A

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

   5. lower-tan.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 98.5

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

10. Applied rewrites 98.5%

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

Alternative 5: 98.1% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (sqrt
  (/
   1.0
   (+
    1.0
    (*
     (/
      (* alphay alphay)
      (pow
       (sin (atan (* (/ alphay alphax) (tan (fma 0.5 PI (* 2.0 (* u1 PI)))))))
       2.0))
     (/ u0 (- 1.0 u0)))))))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.9%

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

   1. lift-/.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in alphax around inf

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

   1. lower-sqrt.f32 N/A

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

10. Applied rewrites 98.1%

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

Alternative 6: 96.5% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    0.5
    (*
     (/
      (* alphay alphay)
      (pow
       (sin (atan (* (/ alphay alphax) (tan (fma 0.5 PI (* 2.0 (* u1 PI)))))))
       2.0))
     (/ u0 (- 1.0 u0)))))))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in alphay around 0

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

   1. lower-+.f32 N/A

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

10. Applied rewrites 96.5%

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

Alternative 7: 96.5% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (+
  1.0
  (*
   -0.5
   (*
    (/
     (* alphay alphay)
     (pow
      (sin (atan (* (/ alphay alphax) (tan (fma 0.5 PI (* 2.0 (* u1 PI)))))))
      2.0))
    (/ u0 (- 1.0 u0))))))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.9%

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

   1. lift-/.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in alphay around 0

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

   1. lower-+.f32 N/A

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

10. Applied rewrites 96.5%

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

Alternative 8: 91.4% accurate, 1436.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay) :precision binary32 1.0)

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(1.0)
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.9%

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

   1. lift-/.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

5. Applied rewrites 99.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 91.4%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025107 
(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))))))