Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 12.3s

Alternatives: 13

Speedup: 1.0×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((single(6.28318530718) * u2));
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((single(6.28318530718) * u2));
end

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (+ (/ 1.0 u1) -1.0) -0.5) (cos (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * cosf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * cos((single(6.28318530718) * u2));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{\frac{u1}{1 - u1}}\right), \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. pow1/2 N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{u1}{1 - u1}\right)}^{\frac{1}{2}}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1}{\frac{1 - u1}{u1}}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   6. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1 - u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} - \frac{u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \cos \left(\frac{314159265359}{50000000000} \cdot \color{blue}{u2}\right)\right) \]

   14. cos-lowering-cos.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   15. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right)} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5950000286102295:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.5950000286102295)
   (*
    (pow (+ (/ 1.0 u1) -1.0) -0.5)
    (+
     1.0
     (*
      u2
      (*
       u2
       (+
        -19.739208802181317
        (*
         u2
         (* u2 (+ 64.93939402268539 (* (* u2 u2) -85.45681720672748)))))))))
   (* (cos (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.5950000286102295f) {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f) * (1.0f + (u2 * (u2 * (-19.739208802181317f + (u2 * (u2 * (64.93939402268539f + ((u2 * u2) * -85.45681720672748f))))))));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 * (1.0f + u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.5950000286102295e0) then
        tmp = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (1.0e0 + (u2 * (u2 * ((-19.739208802181317e0) + (u2 * (u2 * (64.93939402268539e0 + ((u2 * u2) * (-85.45681720672748e0)))))))))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt((u1 * (1.0e0 + u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.5950000286102295))
		tmp = Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(Float32(-19.739208802181317) + Float32(u2 * Float32(u2 * Float32(Float32(64.93939402268539) + Float32(Float32(u2 * u2) * Float32(-85.45681720672748))))))))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.5950000286102295))
		tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (single(1.0) + (u2 * (u2 * (single(-19.739208802181317) + (u2 * (u2 * (single(64.93939402268539) + ((u2 * u2) * single(-85.45681720672748)))))))));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt((u1 * (single(1.0) + u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.595000029

   1. Initial program 99.2%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{\frac{u1}{1 - u1}}\right), \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{u1}{1 - u1}\right)}^{\frac{1}{2}}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1}{\frac{1 - u1}{u1}}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      6. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1 - u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} - \frac{u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \cos \left(\frac{314159265359}{50000000000} \cdot \color{blue}{u2}\right)\right) \]

      14. cos-lowering-cos.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

      15. *-lowering-*.f32 99.2%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right)} \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \left(u2 \cdot \color{blue}{\left(u2 \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \color{blue}{\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \left({u2}^{2} \cdot \color{blue}{\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f32 99.1%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 99.1%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{\left(1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\right)\right)} \]

   if 0.595000029 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      3. +-lowering-+.f32 88.0%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   5. Simplified 88.0%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5950000286102295:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999855160713196:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.9999855160713196)
   (* (+ 1.0 (* u2 (* u2 -19.739208802181317))) (sqrt u1))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9999855160713196f) {
		tmp = (1.0f + (u2 * (u2 * -19.739208802181317f))) * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if (cos((6.28318530718e0 * u2)) <= 0.9999855160713196e0) then
        tmp = (1.0e0 + (u2 * (u2 * (-19.739208802181317e0)))) * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9999855160713196))
		tmp = Float32(Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))) * sqrt(u1));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (cos((single(6.28318530718) * u2)) <= single(0.9999855160713196))
		tmp = (single(1.0) + (u2 * (u2 * single(-19.739208802181317)))) * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999985516

   1. Initial program 98.1%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{\frac{u1}{1 - u1}}\right), \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{u1}{1 - u1}\right)}^{\frac{1}{2}}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1}{\frac{1 - u1}{u1}}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      6. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1 - u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} - \frac{u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \cos \left(\frac{314159265359}{50000000000} \cdot \color{blue}{u2}\right)\right) \]

      14. cos-lowering-cos.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

      15. *-lowering-*.f32 98.3%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   4. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right)} \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \color{blue}{\left({u2}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(u2 \cdot \color{blue}{u2}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 66.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u2}\right)\right)\right)\right) \]

   7. Simplified 66.3%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{u1}\right), \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(u1\right), \left(\color{blue}{1} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(u1\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(u1\right), \mathsf{+.f32}\left(1, \left({u2}^{2} \cdot \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(u1\right), \mathsf{+.f32}\left(1, \left(\left(u2 \cdot u2\right) \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(u1\right), \mathsf{+.f32}\left(1, \left(u2 \cdot \color{blue}{\left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)}\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(u1\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)}\right)\right)\right) \]

      8. *-lowering-*.f32 54.9%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(u1\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}\right)\right)\right)\right) \]

   10. Simplified 54.9%

       \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)} \]

   if 0.999985516 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 - u1}} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      3. rgt-mult-inverse N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}} \]

      7. *-rgt-identity N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}} \]

      8. distribute-lft-in N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      12. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right) \]

      18. distribute-lft-in N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999855160713196:\\ \;\;\;\;\left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999855160713196:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.9999855160713196)
   (* (+ 1.0 (* -19.739208802181317 (* u2 u2))) (sqrt u1))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.9999855160713196f) {
		tmp = (1.0f + (-19.739208802181317f * (u2 * u2))) * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if (cos((6.28318530718e0 * u2)) <= 0.9999855160713196e0) then
        tmp = (1.0e0 + ((-19.739208802181317e0) * (u2 * u2))) * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.9999855160713196))
		tmp = Float32(Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))) * sqrt(u1));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (cos((single(6.28318530718) * u2)) <= single(0.9999855160713196))
		tmp = (single(1.0) + (single(-19.739208802181317) * (u2 * u2))) * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.999985516

   1. Initial program 98.1%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{\frac{u1}{1 - u1}}\right), \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{u1}{1 - u1}\right)}^{\frac{1}{2}}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1}{\frac{1 - u1}{u1}}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      6. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1 - u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} - \frac{u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \cos \left(\frac{314159265359}{50000000000} \cdot \color{blue}{u2}\right)\right) \]

      14. cos-lowering-cos.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

      15. *-lowering-*.f32 98.3%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   4. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right)} \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \color{blue}{\left({u2}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(u2 \cdot \color{blue}{u2}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 66.3%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u2}\right)\right)\right)\right) \]

   7. Simplified 66.3%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(\sqrt{u1}\right)}, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. sqrt-lowering-sqrt.f32 54.9%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(u1\right), \mathsf{+.f32}\left(\color{blue}{1}, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right) \]

   10. Simplified 54.9%

       \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]

   if 0.999985516 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 - u1}} \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      3. rgt-mult-inverse N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}} \]

      7. *-rgt-identity N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}} \]

      8. distribute-lft-in N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      12. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right) \]

      18. distribute-lft-in N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1\right)\right)\right) \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999855160713196:\\ \;\;\;\;\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (cos (* 6.28318530718 u2)) (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((6.28318530718f * u2)) * sqrtf((u1 / (1.0f - u1)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = cos((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = cos((single(6.28318530718) * u2)) * sqrt((u1 / (single(1.0) - u1)));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Final simplification 99.0%

   \[\leadsto \cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
4. Add Preprocessing

Alternative 6: 93.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (pow (+ (/ 1.0 u1) -1.0) -0.5)
  (+
   1.0
   (*
    u2
    (*
     u2
     (+
      -19.739208802181317
      (*
       u2
       (* u2 (+ 64.93939402268539 (* (* u2 u2) -85.45681720672748))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -0.5f) * (1.0f + (u2 * (u2 * (-19.739208802181317f + (u2 * (u2 * (64.93939402268539f + ((u2 * u2) * -85.45681720672748f))))))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (1.0e0 + (u2 * (u2 * ((-19.739208802181317e0) + (u2 * (u2 * (64.93939402268539e0 + ((u2 * u2) * (-85.45681720672748e0)))))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(Float32(-19.739208802181317) + Float32(u2 * Float32(u2 * Float32(Float32(64.93939402268539) + Float32(Float32(u2 * u2) * Float32(-85.45681720672748))))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (single(1.0) + (u2 * (u2 * (single(-19.739208802181317) + (u2 * (u2 * (single(64.93939402268539) + ((u2 * u2) * single(-85.45681720672748)))))))));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{\frac{u1}{1 - u1}}\right), \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. pow1/2 N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{u1}{1 - u1}\right)}^{\frac{1}{2}}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1}{\frac{1 - u1}{u1}}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   6. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1 - u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} - \frac{u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \cos \left(\frac{314159265359}{50000000000} \cdot \color{blue}{u2}\right)\right) \]

   14. cos-lowering-cos.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   15. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right)} \]

5. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \left(u2 \cdot \color{blue}{\left(u2 \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \frac{-98696044010906577398881}{5000000000000000000000}\right)\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \color{blue}{\left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \left({u2}^{2} \cdot \color{blue}{\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f32 94.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 94.0%

   \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{\left(1 + u2 \cdot \left(u2 \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 7: 91.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot 64.93939402268539\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (+
   1.0
   (* (* u2 u2) (+ -19.739208802181317 (* u2 (* u2 64.93939402268539)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * (-19.739208802181317f + (u2 * (u2 * 64.93939402268539f)))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * ((-19.739208802181317e0) + (u2 * (u2 * 64.93939402268539e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(u2 * Float32(u2 * Float32(64.93939402268539)))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * (single(-19.739208802181317) + (u2 * (u2 * single(64.93939402268539))))));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

4. Applied egg-rr 98.9%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{-1 + u1 \cdot \left(u1 \cdot u1\right)}{-\left(1 + u1 \cdot \left(u1 + 1\right)\right)}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\mathsf{neg}\left(\frac{-1 + u1 \cdot \left(u1 \cdot u1\right)}{1 + u1 \cdot \left(u1 + 1\right)}\right)\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   2. distribute-frac-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot \left(u1 \cdot u1\right)\right)\right)}{1 + u1 \cdot \left(u1 + 1\right)}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(u1 \cdot \left(u1 \cdot u1\right)\right)\right)}{1 + u1 \cdot \left(u1 + 1\right)}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{1 + \left(\mathsf{neg}\left(u1 \cdot \left(u1 \cdot u1\right)\right)\right)}{1 + u1 \cdot \left(u1 + 1\right)}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{1 - u1 \cdot \left(u1 \cdot u1\right)}{1 + u1 \cdot \left(u1 + 1\right)}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{{1}^{3} - u1 \cdot \left(u1 \cdot u1\right)}{1 + u1 \cdot \left(u1 + 1\right)}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   7. cube-unmult N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{{1}^{3} - {u1}^{3}}{1 + u1 \cdot \left(u1 + 1\right)}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + u1 \cdot \left(u1 + 1\right)}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   10. flip3-- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(1 - u1\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   11. flip-- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{1 \cdot 1 - u1 \cdot u1}{u1 + 1}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\frac{1}{\frac{u1 + 1}{1 \cdot 1 - u1 \cdot u1}}\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \left(\frac{u1 + 1}{1 \cdot 1 - u1 \cdot u1}\right)\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(u1 + 1\right), \left(1 \cdot 1 - u1 \cdot u1\right)\right)\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(1 + u1\right), \left(1 \cdot 1 - u1 \cdot u1\right)\right)\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   17. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \left(1 \cdot 1 - u1 \cdot u1\right)\right)\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \left(1 - u1 \cdot u1\right)\right)\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   19. --lowering--.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \left(u1 \cdot u1\right)\right)\right)\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   20. *-lowering-*.f32 98.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

6. Applied egg-rr 98.9%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{1}{\frac{1 + u1}{1 - u1 \cdot u1}}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

7. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right) \]
8. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)}\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{-98696044010906577398881}{5000000000000000000000}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\frac{-98696044010906577398881}{5000000000000000000000} + \color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}}\right)\right)\right)\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left({u2}^{2} \cdot \color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(\left(u2 \cdot u2\right) \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(u2 \cdot \color{blue}{\left(u2 \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)}\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 91.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{+.f32}\left(1, u1\right), \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(u1, u1\right)\right)\right)\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right) \]

9. Simplified 91.0%

   \[\leadsto \sqrt{\frac{u1}{\frac{1}{\frac{1 + u1}{1 - u1 \cdot u1}}}} \cdot \color{blue}{\left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot 64.93939402268539\right)\right)\right)} \]
10. Step-by-step derivation

    1. sqrt-div N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\sqrt{u1}}{\sqrt{\frac{1}{\frac{1 + u1}{1 - u1 \cdot u1}}}}\right), \mathsf{+.f32}\left(\color{blue}{1}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

    2. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\sqrt{u1}}{\sqrt{\frac{1 - u1 \cdot u1}{1 + u1}}}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\sqrt{u1}}{\sqrt{\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}}}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

    4. flip-- N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\sqrt{u1}}{\sqrt{1 - u1}}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

    5. sqrt-div N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{\frac{u1}{1 - u1}}\right), \mathsf{+.f32}\left(\color{blue}{1}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

    6. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{u1}{1 - u1}\right)\right), \mathsf{+.f32}\left(\color{blue}{1}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

    7. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(1 - u1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

    8. --lowering--.f32 91.1%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 91.1%

    \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + u2 \cdot \left(u2 \cdot 64.93939402268539\right)\right)\right) \]
12. Add Preprocessing

Alternative 8: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 - u1}{u1}\right)}^{-0.5} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (/ (- 1.0 u1) u1) -0.5) (+ 1.0 (* -19.739208802181317 (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f - u1) / u1), -0.5f) * (1.0f + (-19.739208802181317f * (u2 * u2)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (((1.0e0 - u1) / u1) ** (-0.5e0)) * (1.0e0 + ((-19.739208802181317e0) * (u2 * u2)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) - u1) / u1) ^ Float32(-0.5)) * Float32(Float32(1.0) + Float32(Float32(-19.739208802181317) * Float32(u2 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) - u1) / u1) ^ single(-0.5)) * (single(1.0) + (single(-19.739208802181317) * (u2 * u2)));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{\frac{u1}{1 - u1}}\right), \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. pow1/2 N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{u1}{1 - u1}\right)}^{\frac{1}{2}}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1}{\frac{1 - u1}{u1}}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   6. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1 - u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} - \frac{u1}{u1}\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \left(-1 \cdot \frac{1}{2}\right)\right), \cos \left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \cos \left(\frac{314159265359}{50000000000} \cdot \color{blue}{u2}\right)\right) \]

   14. cos-lowering-cos.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   15. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right)} \]

5. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \color{blue}{\left({u2}^{2}\right)}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \left(u2 \cdot \color{blue}{u2}\right)\right)\right)\right) \]

   4. *-lowering-*.f32 88.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u2}\right)\right)\right)\right) \]

7. Simplified 88.0%

   \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \color{blue}{\left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right)} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\color{blue}{\left(\frac{1 + -1 \cdot u1}{u1}\right)}, \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\left(1 + -1 \cdot u1\right), u1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right), u1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\left(1 - u1\right), u1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right) \]

   4. --lowering--.f32 88.1%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u1\right), u1\right), \frac{-1}{2}\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right) \]

10. Simplified 88.1%

    \[\leadsto {\color{blue}{\left(\frac{1 - u1}{u1}\right)}}^{-0.5} \cdot \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \]
11. Add Preprocessing

Alternative 9: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* u2 (* u2 -19.739208802181317)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (u2 * (u2 * single(-19.739208802181317))));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left({u2}^{2} \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}}\right) \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} + \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   3. distribute-rgt1-in N/A

      \[\leadsto \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2} + 1\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   4. *-commutative N/A

      \[\leadsto \left({u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000} + 1\right) \cdot \sqrt{\frac{\color{blue}{u1}}{1 - u1}} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left({u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000} + 1\right), \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}}\right)}\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left({u2}^{2} \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right), \left(\sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(\left(u2 \cdot u2\right) \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right), \left(\sqrt{\frac{u1}{1 - u1}}\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\left(u2 \cdot \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{\color{blue}{u1}}{1 - u1}}\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \left(u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{\color{blue}{u1}}{1 - u1}}\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1}{1 - u1}}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{1 - u1}}\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   13. rgt-mult-inverse N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}}\right)\right) \]

   17. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}}\right)\right) \]

   18. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}\right)\right) \]

   20. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \frac{-98696044010906577398881}{5000000000000000000000}\right)\right), 1\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}\right)\right) \]

5. Simplified 88.0%

   \[\leadsto \color{blue}{\left(u2 \cdot \left(u2 \cdot -19.739208802181317\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

6. Final simplification 88.0%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right) \]
7. Add Preprocessing

Alternative 10: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (/ u1 (- 1.0 u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1)));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 - u1}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   3. rgt-mult-inverse N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}} \]

   7. *-rgt-identity N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}} \]

   8. distribute-lft-in N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}} \]

   10. sub-neg N/A

       \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}} \]

   11. associate-*r* N/A

       \[\leadsto \sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   12. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

   13. *-rgt-identity N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right) \]

   18. distribute-lft-in N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1\right)\right)\right) \]

5. Simplified 78.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
6. Add Preprocessing

Alternative 11: 71.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (* u1 (+ 1.0 u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + u1)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 * (1.0e0 + u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(1.0) + u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + u1)));
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 - u1}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   3. rgt-mult-inverse N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}} \]

   7. *-rgt-identity N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}} \]

   8. distribute-lft-in N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}} \]

   10. sub-neg N/A

       \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}} \]

   11. associate-*r* N/A

       \[\leadsto \sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   12. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

   13. *-rgt-identity N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right) \]

   18. distribute-lft-in N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1\right)\right)\right) \]

5. Simplified 78.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right) \]

   3. +-lowering-+.f32 69.3%

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right) \]

8. Simplified 69.3%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \]

9. Final simplification 69.3%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
10. Add Preprocessing

Alternative 12: 63.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 - u1}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   3. rgt-mult-inverse N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}} \]

   7. *-rgt-identity N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}} \]

   8. distribute-lft-in N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}} \]

   10. sub-neg N/A

       \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}} \]

   11. associate-*r* N/A

       \[\leadsto \sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   12. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

   13. *-rgt-identity N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right) \]

   18. distribute-lft-in N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1\right)\right)\right) \]

5. Simplified 78.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1}} \]
7. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 61.2%

      \[\leadsto \mathsf{sqrt.f32}\left(u1\right) \]

8. Simplified 61.2%

   \[\leadsto \color{blue}{\sqrt{u1}} \]
9. Add Preprocessing

Alternative 13: 19.2% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 1.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(1.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 - u1}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   3. rgt-mult-inverse N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}} \]

   7. *-rgt-identity N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}} \]

   8. distribute-lft-in N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}} \]

   10. sub-neg N/A

       \[\leadsto \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}} \]

   11. associate-*r* N/A

       \[\leadsto \sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   12. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

   13. *-rgt-identity N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

   17. +-commutative N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right) \]

   18. distribute-lft-in N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1\right)\right)\right) \]

5. Simplified 78.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
6. Step-by-step derivation

   1. sqrt-div N/A

      \[\leadsto \frac{\sqrt{u1}}{\color{blue}{\sqrt{1 - u1}}} \]

   2. flip3-- N/A

      \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{1 - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \]

   4. cube-unmult N/A

      \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{1 - u1 \cdot \left(u1 \cdot u1\right)}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \]

   5. sub-neg N/A

      \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{1 + \left(\mathsf{neg}\left(u1 \cdot \left(u1 \cdot u1\right)\right)\right)}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(u1 \cdot \left(u1 \cdot u1\right)\right)\right)}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \]

   7. distribute-neg-in N/A

      \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot \left(u1 \cdot u1\right)\right)\right)}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot \left(u1 \cdot u1\right)\right)\right)}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{\mathsf{neg}\left(\left(-1 + u1 \cdot \left(u1 \cdot u1\right)\right)\right)}{1 + u1 \cdot \left(u1 + 1\right)}}} \]

   10. distribute-frac-neg N/A

       \[\leadsto \frac{\sqrt{u1}}{\sqrt{\mathsf{neg}\left(\frac{-1 + u1 \cdot \left(u1 \cdot u1\right)}{1 + u1 \cdot \left(u1 + 1\right)}\right)}} \]

   11. distribute-frac-neg2 N/A

       \[\leadsto \frac{\sqrt{u1}}{\sqrt{\frac{-1 + u1 \cdot \left(u1 \cdot u1\right)}{\mathsf{neg}\left(\left(1 + u1 \cdot \left(u1 + 1\right)\right)\right)}}} \]

   12. sqrt-div N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{-1 + u1 \cdot \left(u1 \cdot u1\right)}{\mathsf{neg}\left(\left(1 + u1 \cdot \left(u1 + 1\right)\right)\right)}}} \]

   13. pow1/2 N/A

       \[\leadsto {\left(\frac{u1}{\frac{-1 + u1 \cdot \left(u1 \cdot u1\right)}{\mathsf{neg}\left(\left(1 + u1 \cdot \left(u1 + 1\right)\right)\right)}}\right)}^{\color{blue}{\frac{1}{2}}} \]

   14. sqr-pow N/A

       \[\leadsto {\left(\frac{u1}{\frac{-1 + u1 \cdot \left(u1 \cdot u1\right)}{\mathsf{neg}\left(\left(1 + u1 \cdot \left(u1 + 1\right)\right)\right)}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \color{blue}{{\left(\frac{u1}{\frac{-1 + u1 \cdot \left(u1 \cdot u1\right)}{\mathsf{neg}\left(\left(1 + u1 \cdot \left(u1 + 1\right)\right)\right)}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

7. Applied egg-rr 78.2%

   \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1} \cdot \frac{u1}{1 - u1}\right)}^{0.25}} \]

8. Taylor expanded in u1 around -inf

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

10. Simplified 19.1%

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

Reproduce

herbie shell --seed 2024149 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))