Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 12.7s

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: 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

Derivation

1. Initial program 99.1%

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

Alternative 2: 93.5% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (* u1 (+ (/ 1.0 u1) -1.0))))
  (+
   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 sqrtf((u1 / (u1 * ((1.0f / u1) + -1.0f)))) * (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 = sqrt((u1 / (u1 * ((1.0e0 / u1) + (-1.0e0))))) * (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(sqrt(Float32(u1 / Float32(u1 * Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * u2) * Float32(Float32(64.93939402268539) + Float32(u2 * Float32(u2 * Float32(-85.45681720672748)))))))))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.1%

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

5. Taylor expanded in u1 around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. /-lowering-/.f32 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in u2 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-lowering-*.f32 95.8%

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

10. Simplified 95.8%

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

Alternative 3: 91.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (u1 * ((1.0f / u1) + -1.0f)))) * (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 / (u1 * ((1.0e0 / u1) + (-1.0e0))))) * (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(u1 * Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * u2) * Float32(64.93939402268539))))))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.1%

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

5. Taylor expanded in u1 around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. /-lowering-/.f32 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in u2 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f32 94.2%

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

10. Simplified 94.2%

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

Alternative 4: 88.7% accurate, 1.8× speedup

Math

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

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(Float32(u2 * 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 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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. *-lowering-*.f32 N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. *-rgt-identity N/A

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

   10. sub-neg N/A

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

   11. rgt-mult-inverse N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. *-rgt-identity N/A

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

   16. distribute-lft-in N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. associate-*r* N/A

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

5. Simplified 91.4%

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

6. Final simplification 91.4%

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

Alternative 5: 80.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ {\left(t\_0 \cdot t\_0\right)}^{0.25} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1)))) (pow (* t_0 t_0) 0.25)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	return powf((t_0 * t_0), 0.25f);
}

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) :: t_0
    t_0 = u1 / (1.0e0 - u1)
    code = (t_0 * t_0) ** 0.25e0
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	return Float32(t_0 * t_0) ^ Float32(0.25)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = u1 / (single(1.0) - u1);
	tmp = (t_0 * t_0) ^ single(0.25);
end

Derivation

1. Initial program 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 83.9%

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

7. Applied egg-rr 84.0%

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

Alternative 6: 80.4% 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 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 83.9%

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

Alternative 7: 72.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (u1 + 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 = sqrt((u1 * (u1 + 1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 83.9%

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

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 76.3%

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

8. Simplified 76.3%

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

Alternative 8: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(u1 \cdot u1\right)}^{0.25} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (pow (* u1 u1) 0.25))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((u1 * u1), 0.25f);
}

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 = (u1 * u1) ** 0.25e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 * u1) ^ Float32(0.25)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u1 * u1) ^ single(0.25);
end

Derivation

1. Initial program 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 83.9%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 68.1%

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

8. Simplified 68.1%

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

   1. pow1/2 N/A

      \[\leadsto {u1}^{\color{blue}{\frac{1}{2}}} \]

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

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

      \[\leadsto \mathsf{pow.f32}\left(\left(u1 \cdot u1\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

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

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{*.f32}\left(u1, u1\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

   6. metadata-eval 68.1%

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{*.f32}\left(u1, u1\right), \frac{1}{4}\right) \]

10. Applied egg-rr 68.1%

    \[\leadsto \color{blue}{{\left(u1 \cdot u1\right)}^{0.25}} \]
11. Add Preprocessing

Alternative 9: 63.7% 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 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 83.9%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 68.1%

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

8. Simplified 68.1%

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

Alternative 10: 19.8% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (+
  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 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 + ((u2 * u2) * ((-19.739208802181317e0) + ((u2 * u2) * (64.93939402268539e0 + (u2 * (u2 * (-85.45681720672748e0)))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 74.7%

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

5. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   2. *-lowering-*.f32 19.6%

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

7. Simplified 19.6%

   \[\leadsto \color{blue}{\cos \left(6.28318530718 \cdot u2\right)} \]

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)} \]
9. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

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

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

   17. *-lowering-*.f32 19.5%

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

10. Simplified 19.5%

    \[\leadsto \color{blue}{1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot \left(64.93939402268539 + u2 \cdot \left(u2 \cdot -85.45681720672748\right)\right)\right)} \]
11. Add Preprocessing

Alternative 11: 19.7% accurate, 16.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = 1.0e0 + ((u2 * u2) * ((-19.739208802181317e0) + ((u2 * u2) * 64.93939402268539e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 74.7%

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

5. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   2. *-lowering-*.f32 19.6%

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

7. Simplified 19.6%

   \[\leadsto \color{blue}{\cos \left(6.28318530718 \cdot u2\right)} \]

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)} \]
9. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f32 19.4%

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

10. Simplified 19.4%

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

Alternative 12: 19.6% accurate, 29.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = 1.0e0 + ((u2 * u2) * (-19.739208802181317e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 74.7%

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

5. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   2. *-lowering-*.f32 19.6%

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

7. Simplified 19.6%

   \[\leadsto \color{blue}{\cos \left(6.28318530718 \cdot u2\right)} \]

8. Taylor expanded in u2 around 0

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

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

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 19.4%

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

10. Simplified 19.4%

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

11. Final simplification 19.4%

    \[\leadsto 1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317 \]
12. 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 99.1%

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

4. Applied egg-rr 74.7%

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

5. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{cos.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right) \]

   2. *-lowering-*.f32 19.6%

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

7. Simplified 19.6%

   \[\leadsto \color{blue}{\cos \left(6.28318530718 \cdot u2\right)} \]

8. Taylor expanded in u2 around 0

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

10. Simplified 19.3%

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

Reproduce

herbie shell --seed 2024155 
(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))))