Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 12.4s

Alternatives: 18

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 99.0%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.800000011920929:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.800000011920929)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (+
     1.0
     (*
      (* u2 u2)
      (+
       -19.739208802181317
       (* (* u2 u2) (+ 64.93939402268539 (* (* u2 u2) -85.45681720672748)))))))
   (* (cos (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.800000011920929f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * (-19.739208802181317f + ((u2 * u2) * (64.93939402268539f + ((u2 * u2) * -85.45681720672748f))))));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	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) <= 0.800000011920929e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * ((-19.739208802181317e0) + ((u2 * u2) * (64.93939402268539e0 + ((u2 * u2) * (-85.45681720672748e0)))))))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.800000011920929))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * 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(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.800000011920929))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (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 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.800000012

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 99.3%

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

   5. Simplified 99.3%

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

   if 0.800000012 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 84.3%

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

   5. Simplified 84.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.800000011920929:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 1:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 1.0)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (+
     1.0
     (*
      (* u2 u2)
      (+
       -19.739208802181317
       (* (* u2 u2) (+ 64.93939402268539 (* (* u2 u2) -85.45681720672748)))))))
   (* (cos (* 6.28318530718 u2)) (pow (* u1 u1) 0.25))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 1.0f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * (-19.739208802181317f + ((u2 * u2) * (64.93939402268539f + ((u2 * u2) * -85.45681720672748f))))));
	} else {
		tmp = cosf((6.28318530718f * u2)) * powf((u1 * u1), 0.25f);
	}
	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) <= 1.0e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * ((-19.739208802181317e0) + ((u2 * u2) * (64.93939402268539e0 + ((u2 * u2) * (-85.45681720672748e0)))))))
    else
        tmp = cos((6.28318530718e0 * u2)) * ((u1 * u1) ** 0.25e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(1.0))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * u2) * Float32(Float32(64.93939402268539) + Float32(Float32(u2 * u2) * Float32(-85.45681720672748))))))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * (Float32(u1 * u1) ^ Float32(0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(1.0))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (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)) * ((u1 * u1) ^ single(0.25));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 99.1%

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

   5. Simplified 99.1%

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

   if 1 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.4%

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

   3. Taylor expanded in u1 around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f32 71.2%

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

   5. Simplified 71.2%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

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

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

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

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

      6. metadata-eval 71.4%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{pow.f32}\left(\mathsf{*.f32}\left(u1, u1\right), \frac{1}{4}\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

   7. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{{\left(u1 \cdot u1\right)}^{0.25}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 1:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot \left(-19.739208802181317 + \left(u2 \cdot u2\right) \cdot \left(64.93939402268539 + \left(u2 \cdot u2\right) \cdot -85.45681720672748\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 1.0f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * (-19.739208802181317f + ((u2 * u2) * (64.93939402268539f + ((u2 * u2) * -85.45681720672748f))))));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(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) <= 1.0e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * ((-19.739208802181317e0) + ((u2 * u2) * (64.93939402268539e0 + ((u2 * u2) * (-85.45681720672748e0)))))))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt(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(1.0))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * u2) * Float32(Float32(64.93939402268539) + Float32(Float32(u2 * u2) * Float32(-85.45681720672748))))))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f32 99.1%

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

   5. Simplified 99.1%

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

   if 1 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.4%

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

   3. Taylor expanded in u1 around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f32 71.2%

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

   5. Simplified 71.2%

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

4. Final simplification 96.8%

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

Alternative 5: 93.8% accurate, 1.7× speedup

Math

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

FPCore

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

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 + ((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 / (1.0e0 - u1))) * (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(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(Float32(-19.739208802181317) + Float32(Float32(u2 * u2) * Float32(Float32(64.93939402268539) + Float32(Float32(u2 * u2) * Float32(-85.45681720672748))))))))
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) + ((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. Taylor expanded in u2 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 93.4%

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

5. Simplified 93.4%

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

Alternative 6: 85.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.00107999995816499:\\ \;\;\;\;\sqrt{u1 \cdot \left(u1 + 1\right)} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.00107999995816499)
     (* (sqrt (* u1 (+ u1 1.0))) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
     (sqrt t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.00107999995816499f) {
		tmp = sqrtf((u1 * (u1 + 1.0f))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} else {
		tmp = sqrtf(t_0);
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = u1 / (1.0e0 - u1)
    if (t_0 <= 0.00107999995816499e0) then
        tmp = sqrt((u1 * (u1 + 1.0e0))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = sqrt(t_0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00107999995816499))
		tmp = Float32(sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	else
		tmp = sqrt(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u1 / (single(1.0) - u1);
	tmp = single(0.0);
	if (t_0 <= single(0.00107999995816499))
		tmp = sqrt((u1 * (u1 + single(1.0)))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	else
		tmp = sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00107999996

   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-rgt-identity N/A

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

      13. sub-neg N/A

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

      14. rgt-mult-inverse N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-frac2 N/A

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

      17. mul-1-neg N/A

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

      18. *-rgt-identity N/A

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

      19. distribute-lft-in N/A

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

   5. Simplified 88.6%

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

   6. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 88.2%

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

   8. Simplified 88.2%

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

   if 0.00107999996 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 78.4%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.00107999995816499:\\ \;\;\;\;\sqrt{u1 \cdot \left(u1 + 1\right)} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% 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

3. Taylor expanded in u2 around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f32 91.9%

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

5. Simplified 91.9%

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

Alternative 8: 83.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (sqrt u1) (+ 1.0 (* u2 (* u2 -19.739208802181317))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
	}
	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) <= 0.009999999776482582e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(u1) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.009999999776482582))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.009999999776482582))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(u1) * (single(1.0) + (u2 * (u2 * single(-19.739208802181317))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 99.6%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 96.7%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-rgt-identity N/A

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

      13. sub-neg N/A

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

      14. rgt-mult-inverse N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-frac2 N/A

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

      17. mul-1-neg N/A

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

      18. *-rgt-identity N/A

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

      19. distribute-lft-in N/A

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

   5. Simplified 63.0%

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 63.0%

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

   7. Applied egg-rr 63.0%

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

   8. Taylor expanded in u1 around 0

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

   10. Simplified 52.1%

       \[\leadsto \left(1 + \left(u2 \cdot -19.739208802181317\right) \cdot u2\right) \cdot \sqrt{\color{blue}{u1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (sqrt (/ u1 (- 1.0 u1)))
   (* (sqrt u1) (+ 1.0 (* (* u2 u2) -19.739208802181317)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	}
	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) <= 0.009999999776482582e0) then
        tmp = sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(u1) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.009999999776482582))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.009999999776482582))
		tmp = sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(u1) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 99.6%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 96.7%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-rgt-identity N/A

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

      13. sub-neg N/A

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

      14. rgt-mult-inverse N/A

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

      15. mul-1-neg N/A

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

      16. distribute-neg-frac2 N/A

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

      17. mul-1-neg N/A

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

      18. *-rgt-identity N/A

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

      19. distribute-lft-in N/A

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

   5. Simplified 63.0%

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

   6. Taylor expanded in u1 around 0

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

   8. Simplified 52.1%

      \[\leadsto \left(1 + -19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot \sqrt{\color{blue}{u1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.5% 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. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt1-in N/A

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

   6. +-commutative N/A

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

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

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

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

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

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

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

   10. unpow2 N/A

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

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

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

   12. *-rgt-identity N/A

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

   13. sub-neg N/A

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

   14. rgt-mult-inverse N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-frac2 N/A

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

   17. mul-1-neg N/A

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

   18. *-rgt-identity N/A

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

   19. distribute-lft-in N/A

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

5. Simplified 88.0%

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

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 88.0%

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

7. Applied egg-rr 88.0%

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

8. Final simplification 88.0%

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

Alternative 11: 88.5% 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 99.0%

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt1-in N/A

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

   6. +-commutative N/A

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

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

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

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

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

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

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

   10. unpow2 N/A

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

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

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

   12. *-rgt-identity N/A

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

   13. sub-neg N/A

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

   14. rgt-mult-inverse N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-frac2 N/A

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

   17. mul-1-neg N/A

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

   18. *-rgt-identity N/A

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

   19. distribute-lft-in N/A

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

5. Simplified 88.0%

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

6. Final simplification 88.0%

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

Alternative 12: 79.9% 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 79.2%

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

Alternative 13: 71.7% 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 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 79.2%

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

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 70.2%

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

8. Simplified 70.2%

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

Alternative 14: 63.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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) ** (-0.5e0)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(1.0) / u1) ^ single(-0.5);
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 79.2%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 62.3%

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

8. Simplified 62.3%

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

   1. pow1/2 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

      \[\leadsto {\left(\frac{1}{u1}\right)}^{\frac{-1}{2}} \]

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

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

   6. /-lowering-/.f32 62.3%

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

10. Applied egg-rr 62.3%

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

Alternative 15: 63.5% 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 79.2%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 62.3%

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

8. Simplified 62.3%

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

Alternative 16: 19.7% accurate, 16.1× speedup

Math

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

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(u2 * Float32(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 99.0%

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

4. Applied egg-rr 93.2%

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

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 20.1%

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

7. Simplified 20.1%

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

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. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

   13. *-lowering-*.f32 19.8%

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

10. Simplified 19.8%

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

Alternative 17: 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 99.0%

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

4. Applied egg-rr 93.2%

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

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 20.1%

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

7. Simplified 20.1%

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

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 19.5%

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

10. Simplified 19.5%

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

11. Final simplification 19.5%

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

Alternative 18: 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

4. Applied egg-rr 93.2%

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

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 20.1%

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

7. Simplified 20.1%

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

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 2024163 
(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))))