Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.3% → 99.9%

Time: 56.3s

Alternatives: 9

Speedup: 2.1×

Specification

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (let* ((t_0
        (atan
         (*
          (/ alphay alphax)
          (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
       (t_1 (sin t_0))
       (t_2 (cos t_0)))
  (/
   1.0
   (sqrt
    (+
     1.0
     (/
      (*
       (/
        1.0
        (+
         (/ (* t_2 t_2) (* alphax alphax))
         (/ (* t_1 t_1) (* alphay alphay))))
       u0)
      (- 1.0 u0)))))))

C

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

Julia

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

MATLAB

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

Initial Program: 99.3% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (let* ((t_0
        (atan
         (*
          (/ alphay alphax)
          (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
       (t_1 (sin t_0))
       (t_2 (cos t_0)))
  (/
   1.0
   (sqrt
    (+
     1.0
     (/
      (*
       (/
        1.0
        (+
         (/ (* t_2 t_2) (* alphax alphax))
         (/ (* t_1 t_1) (* alphay alphay))))
       u0)
      (- 1.0 u0)))))))

C

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 2.0× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (let* ((t_0
        (asinh (* (tan (* PI (fma u1 2.0 0.5))) (/ alphay alphax)))))
  (pow
   (-
    (/
     u0
     (*
      (- 1.0 u0)
      (+
       (pow (/ (tanh t_0) alphay) 2.0)
       (/ (pow (cosh t_0) -2.0) (* alphax alphax)))))
    -1.0)
   -0.5)))

C

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

Julia

function code(u0, u1, alphax, alphay)
	t_0 = asinh(Float32(tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))) * Float32(alphay / alphax)))
	return Float32(Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32((Float32(tanh(t_0) / alphay) ^ Float32(2.0)) + Float32((cosh(t_0) ^ Float32(-2.0)) / Float32(alphax * alphax))))) - 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%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.9%

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.4% accurate, 2.1× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (let* ((t_0
        (asinh (* (tan (* PI (fma u1 2.0 0.5))) (/ alphay alphax)))))
  (/
   1.0
   (sqrt
    (/
     (+
      (/
       u0
       (+
        (pow (/ (tanh t_0) alphay) 2.0)
        (pow (/ 1.0 (* (cosh t_0) alphax)) 2.0)))
      (- 1.0 u0))
     (- 1.0 u0))))))

C

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

Julia

function code(u0, u1, alphax, alphay)
	t_0 = asinh(Float32(tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))) * Float32(alphay / alphax)))
	return Float32(Float32(1.0) / sqrt(Float32(Float32(Float32(u0 / Float32((Float32(tanh(t_0) / alphay) ^ Float32(2.0)) + (Float32(Float32(1.0) / Float32(cosh(t_0) * alphax)) ^ Float32(2.0)))) + Float32(Float32(1.0) - 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 99.4%

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

5. Applied rewrites 99.3%

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

Alternative 3: 99.4% accurate, 2.1× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (let* ((t_0
        (asinh (* (tan (* PI (fma u1 2.0 0.5))) (/ alphay alphax)))))
  (/
   1.0
   (sqrt
    (fma
     (/
      (* alphay alphay)
      (+
       (pow (/ alphay (* (cosh t_0) alphax)) 2.0)
       (pow (tanh t_0) 2.0)))
     (/ u0 (- 1.0 u0))
     1.0)))))

C

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

Julia

function code(u0, u1, alphax, alphay)
	t_0 = asinh(Float32(tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))) * Float32(alphay / alphax)))
	return Float32(Float32(1.0) / sqrt(fma(Float32(Float32(alphay * alphay) / Float32((Float32(alphay / Float32(cosh(t_0) * alphax)) ^ Float32(2.0)) + (tanh(t_0) ^ Float32(2.0)))), Float32(u0 / Float32(Float32(1.0) - u0)), Float32(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 99.4%

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

5. Applied rewrites 99.4%

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

Alternative 4: 99.3% accurate, 2.1× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (let* ((t_0
        (asinh (* (tan (* PI (fma u1 2.0 0.5))) (/ alphay alphax)))))
  (/
   1.0
   (sqrt
    (-
     (/
      u0
      (*
       (- 1.0 u0)
       (+
        (pow (/ (tanh t_0) alphay) 2.0)
        (/ (pow (cosh t_0) -2.0) (* alphax alphax)))))
     -1.0)))))

C

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

Julia

function code(u0, u1, alphax, alphay)
	t_0 = asinh(Float32(tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))) * Float32(alphay / alphax)))
	return Float32(Float32(1.0) / sqrt(Float32(Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32((Float32(tanh(t_0) / alphay) ^ Float32(2.0)) + Float32((cosh(t_0) ^ Float32(-2.0)) / Float32(alphax * alphax))))) - Float32(-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 99.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.4%

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

5. Applied rewrites 99.4%

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

Alternative 5: 97.7% accurate, 2.2× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (let* ((t_0 (fma 0.5 PI (* 2.0 (* u1 PI)))))
  (/
   1.0
   (sqrt
    (+
     1.0
     (/
      (* (pow alphay 2.0) u0)
      (*
       (pow
        (sin (atan (/ (* alphay (sin t_0)) (* alphax (cos t_0)))))
        2.0)
       (- 1.0 u0))))))))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

Alternative 6: 96.4% accurate, 4.1× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (fma
 (* alphay alphay)
 (*
  (/
   u0
   (*
    (pow
     (tanh
      (asinh (* (tan (* (fma u1 2.0 0.5) PI)) (/ alphay alphax))))
     2.0)
    (- 1.0 u0)))
  -0.5)
 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in alphay around 0

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

   1. lower-+.f32 N/A

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

4. Applied rewrites 96.4%

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

6. Applied rewrites 96.4%

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

Alternative 7: 96.4% accurate, 4.3× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (fma
 (* alphay alphay)
 (*
  (/
   u0
   (*
    (pow (tanh (asinh (* (tan (* 0.5 PI)) (/ alphay alphax)))) 2.0)
    (- 1.0 u0)))
  -0.5)
 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in alphay around 0

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

   1. lower-+.f32 N/A

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

4. Applied rewrites 96.4%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 97.0%

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

7. Applied rewrites 97.0%

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

8. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 96.4%

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

10. Applied rewrites 96.4%

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

    1. lift-+.f32 N/A

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

    2. +-commutative N/A

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

12. Applied rewrites 96.4%

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

Alternative 8: 91.3% accurate, 14.7× speedup

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

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (pow (* (- 1.0 u0) (/ -1.0 (- u0 1.0))) -0.5))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = ((single(1.0) - u0) * (single(-1.0) / (u0 - single(1.0)))) ^ single(-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%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.9%

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in alphax around 0

   \[\leadsto {\left(\color{blue}{\left(1 - u0\right)} \cdot \frac{-1}{u0 - 1}\right)}^{-0.5} \]
9. Step-by-step derivation

   1. lower--.f32 91.3%

      \[\leadsto {\left(\left(1 - \color{blue}{u0}\right) \cdot \frac{-1}{u0 - 1}\right)}^{-0.5} \]

10. Applied rewrites 91.3%

    \[\leadsto {\left(\color{blue}{\left(1 - u0\right)} \cdot \frac{-1}{u0 - 1}\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 9: 88.1% accurate, 27.9× speedup

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

Math

\[\mathsf{fma}\left(alphax \cdot alphax, \frac{u0}{1} \cdot -0.5, 1\right) \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0))
               (and (<= 2.328306437e-10 u1) (<= u1 0.5)))
          (and (<= 0.0001 alphax) (<= alphax 1.0)))
     (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (fma (* alphax alphax) (* (/ u0 1.0) -0.5) 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in alphax around 0

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

   1. lower-+.f32 N/A

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

4. Applied rewrites 41.1%

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

6. Applied rewrites 32.1%

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

7. Taylor expanded in alphax around inf

   \[\leadsto \mathsf{fma}\left(alphax \cdot alphax, \frac{u0}{1 - u0} \cdot -0.5, 1\right) \]
8. Step-by-step derivation

   1. lower--.f32 87.9%

      \[\leadsto \mathsf{fma}\left(alphax \cdot alphax, \frac{u0}{1 - u0} \cdot -0.5, 1\right) \]

9. Applied rewrites 87.9%

   \[\leadsto \mathsf{fma}\left(alphax \cdot alphax, \frac{u0}{1 - u0} \cdot -0.5, 1\right) \]

10. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{fma}\left(alphax \cdot alphax, \frac{u0}{1} \cdot -0.5, 1\right) \]
11. Step-by-step derivation

12. Applied rewrites 88.1%

    \[\leadsto \mathsf{fma}\left(alphax \cdot alphax, \frac{u0}{1} \cdot -0.5, 1\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025356 
(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))))))