Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 11.1s

Alternatives: 16

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

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((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))) * sin((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((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))) * sin((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u1 around inf 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. times-frac 98.3%

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

   3. sub-neg 98.3%

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

   4. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.4%

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

   2. lft-mult-inverse 98.4%

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

7. Simplified 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. inv-pow 98.4%

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

   3. sqrt-pow1 98.4%

      \[\leadsto \left(1 \cdot \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-1}{2}\right)}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. metadata-eval 98.4%

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

9. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(1 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
10. Step-by-step derivation

    1. *-lft-identity 98.4%

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

11. Simplified 98.4%

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

Alternative 2: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.10000000149011612:\\ \;\;\;\;u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.10000000149011612)
   (*
    u2
    (*
     (pow (+ (/ 1.0 u1) -1.0) -0.5)
     (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.10000000149011612f) {
		tmp = u2 * (powf(((1.0f / u1) + -1.0f), -0.5f) * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (1.0f + u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.10000000149011612e0) then
        tmp = u2 * ((((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (1.0e0 + u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.10000000149011612))
		tmp = Float32(u2 * Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.10000000149011612))
		tmp = u2 * ((((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (single(1.0) + u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.100000001

   1. Initial program 98.5%

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

   3. Taylor expanded in u1 around inf 98.6%

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

      1. *-un-lft-identity 98.6%

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

      2. times-frac 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.6%

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

      2. lft-mult-inverse 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in u2 around 0 98.3%

      \[\leadsto \color{blue}{u2 \cdot \left(-41.341702240407926 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 98.3%

         \[\leadsto u2 \cdot \left(\color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) \]

      2. distribute-rgt-out 98.3%

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

      3. sub-neg 98.3%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      4. metadata-eval 98.3%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      5. unpow-1 98.3%

         \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      6. metadata-eval 98.3%

         \[\leadsto u2 \cdot \left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      7. pow-sqr 98.4%

         \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      8. rem-sqrt-square 98.4%

         \[\leadsto u2 \cdot \left(\color{blue}{\left|{\left(\frac{1}{u1} + -1\right)}^{-0.5}\right|} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      9. rem-square-sqrt 97.8%

         \[\leadsto u2 \cdot \left(\left|\color{blue}{\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}}\right| \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      10. fabs-sqr 97.8%

          \[\leadsto u2 \cdot \left(\color{blue}{\left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}\right)} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      11. rem-square-sqrt 98.4%

          \[\leadsto u2 \cdot \left(\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   10. Simplified 98.4%

       \[\leadsto \color{blue}{u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]
   11. Step-by-step derivation

       1. unpow2 98.4%

          \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 6.28318530718\right)\right) \]

   12. Applied egg-rr 98.4%

       \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 6.28318530718\right)\right) \]

   if 0.100000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u1 around 0 90.1%

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

      1. +-commutative 90.1%

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

   5. Simplified 90.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.10000000149011612:\\ \;\;\;\;u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3700000047683716:\\ \;\;\;\;u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.3700000047683716)
   (*
    u2
    (*
     (pow (+ (/ 1.0 u1) -1.0) -0.5)
     (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sin (* 6.28318530718 u2)) (pow (/ 1.0 u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.3700000047683716f) {
		tmp = u2 * (powf(((1.0f / u1) + -1.0f), -0.5f) * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * powf((1.0f / u1), -0.5f);
	}
	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) <= 0.3700000047683716e0) then
        tmp = u2 * ((((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    else
        tmp = sin((6.28318530718e0 * u2)) * ((1.0e0 / u1) ** (-0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.3700000047683716))
		tmp = Float32(u2 * Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * (Float32(Float32(1.0) / u1) ^ Float32(-0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.3700000047683716))
		tmp = u2 * ((((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = sin((single(6.28318530718) * u2)) * ((single(1.0) / u1) ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.370000005

   1. Initial program 98.6%

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

   3. Taylor expanded in u1 around inf 98.7%

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

      1. *-un-lft-identity 98.7%

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

      2. times-frac 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.7%

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

      2. lft-mult-inverse 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in u2 around 0 96.8%

      \[\leadsto \color{blue}{u2 \cdot \left(-41.341702240407926 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 96.8%

         \[\leadsto u2 \cdot \left(\color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) \]

      2. distribute-rgt-out 96.8%

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

      3. sub-neg 96.8%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      4. metadata-eval 96.8%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      5. unpow-1 96.8%

         \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      6. metadata-eval 96.8%

         \[\leadsto u2 \cdot \left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      7. pow-sqr 96.9%

         \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      8. rem-sqrt-square 96.9%

         \[\leadsto u2 \cdot \left(\color{blue}{\left|{\left(\frac{1}{u1} + -1\right)}^{-0.5}\right|} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      9. rem-square-sqrt 96.3%

         \[\leadsto u2 \cdot \left(\left|\color{blue}{\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}}\right| \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      10. fabs-sqr 96.3%

          \[\leadsto u2 \cdot \left(\color{blue}{\left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}\right)} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      11. rem-square-sqrt 96.9%

          \[\leadsto u2 \cdot \left(\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   10. Simplified 96.9%

       \[\leadsto \color{blue}{u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]
   11. Step-by-step derivation

       1. unpow2 96.9%

          \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 6.28318530718\right)\right) \]

   12. Applied egg-rr 96.9%

       \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 6.28318530718\right)\right) \]

   if 0.370000005 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.0%

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

   3. Taylor expanded in u1 around inf 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. times-frac 96.8%

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

      3. sub-neg 96.8%

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

      4. metadata-eval 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. associate-*r/ 97.0%

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

      2. lft-mult-inverse 96.9%

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

   7. Simplified 96.9%

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

      1. *-un-lft-identity 96.9%

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

      2. inv-pow 96.9%

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

      3. sqrt-pow1 96.9%

         \[\leadsto \left(1 \cdot \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-1}{2}\right)}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      4. metadata-eval 96.9%

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

   9. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{\left(1 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   10. Step-by-step derivation

       1. *-lft-identity 96.9%

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

   11. Simplified 96.9%

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

   12. Taylor expanded in u1 around 0 84.5%

       \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3700000047683716:\\ \;\;\;\;u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3700000047683716:\\ \;\;\;\;u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.3700000047683716)
   (*
    u2
    (*
     (pow (+ (/ 1.0 u1) -1.0) -0.5)
     (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.3700000047683716f) {
		tmp = u2 * (powf(((1.0f / u1) + -1.0f), -0.5f) * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	} else {
		tmp = sinf((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) <= 0.3700000047683716e0) then
        tmp = u2 * ((((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    else
        tmp = sin((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(0.3700000047683716))
		tmp = Float32(u2 * Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	else
		tmp = Float32(sin(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(0.3700000047683716))
		tmp = u2 * ((((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	else
		tmp = sin((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) < 0.370000005

   1. Initial program 98.6%

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

   3. Taylor expanded in u1 around inf 98.7%

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

      1. *-un-lft-identity 98.7%

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

      2. times-frac 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.7%

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

      2. lft-mult-inverse 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in u2 around 0 96.8%

      \[\leadsto \color{blue}{u2 \cdot \left(-41.341702240407926 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 96.8%

         \[\leadsto u2 \cdot \left(\color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) \]

      2. distribute-rgt-out 96.8%

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

      3. sub-neg 96.8%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      4. metadata-eval 96.8%

         \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      5. unpow-1 96.8%

         \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      6. metadata-eval 96.8%

         \[\leadsto u2 \cdot \left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      7. pow-sqr 96.9%

         \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      8. rem-sqrt-square 96.9%

         \[\leadsto u2 \cdot \left(\color{blue}{\left|{\left(\frac{1}{u1} + -1\right)}^{-0.5}\right|} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      9. rem-square-sqrt 96.3%

         \[\leadsto u2 \cdot \left(\left|\color{blue}{\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}}\right| \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      10. fabs-sqr 96.3%

          \[\leadsto u2 \cdot \left(\color{blue}{\left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}\right)} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

      11. rem-square-sqrt 96.9%

          \[\leadsto u2 \cdot \left(\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   10. Simplified 96.9%

       \[\leadsto \color{blue}{u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]
   11. Step-by-step derivation

       1. unpow2 96.9%

          \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 6.28318530718\right)\right) \]

   12. Applied egg-rr 96.9%

       \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 6.28318530718\right)\right) \]

   if 0.370000005 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.0%

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

   3. Taylor expanded in u1 around 0 84.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3700000047683716:\\ \;\;\;\;u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) / sqrtf(((1.0f / 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 = sin((6.28318530718e0 * u2)) / sqrt(((1.0e0 / u1) + (-1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u1 around inf 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. times-frac 98.3%

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

   3. sub-neg 98.3%

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

   4. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.4%

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

   2. lft-mult-inverse 98.4%

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

7. Simplified 98.4%

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

   1. *-commutative 98.4%

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

   2. sqrt-div 98.3%

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

   3. metadata-eval 98.3%

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

   4. un-div-inv 98.4%

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

9. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1} + -1}}} \]
10. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 7: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (pow (+ (/ 1.0 u1) -1.0) -0.5)
   (+ 6.28318530718 (* -41.341702240407926 (* u2 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (powf(((1.0f / u1) + -1.0f), -0.5f) * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u2 * ((((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * ((((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
end

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u1 around inf 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. times-frac 98.3%

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

   3. sub-neg 98.3%

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

   4. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.4%

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

   2. lft-mult-inverse 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in u2 around 0 88.5%

   \[\leadsto \color{blue}{u2 \cdot \left(-41.341702240407926 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 88.5%

      \[\leadsto u2 \cdot \left(\color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) \]

   2. distribute-rgt-out 88.5%

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

   3. sub-neg 88.5%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   4. metadata-eval 88.5%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   5. unpow-1 88.5%

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   6. metadata-eval 88.5%

      \[\leadsto u2 \cdot \left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   7. pow-sqr 88.6%

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   8. rem-sqrt-square 88.6%

      \[\leadsto u2 \cdot \left(\color{blue}{\left|{\left(\frac{1}{u1} + -1\right)}^{-0.5}\right|} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   9. rem-square-sqrt 88.1%

      \[\leadsto u2 \cdot \left(\left|\color{blue}{\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}}\right| \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   10. fabs-sqr 88.1%

       \[\leadsto u2 \cdot \left(\color{blue}{\left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}\right)} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   11. rem-square-sqrt 88.6%

       \[\leadsto u2 \cdot \left(\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

10. Simplified 88.6%

    \[\leadsto \color{blue}{u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]
11. Step-by-step derivation

    1. unpow2 88.6%

       \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 6.28318530718\right)\right) \]

12. Applied egg-rr 88.6%

    \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 6.28318530718\right)\right) \]

13. Final simplification 88.6%

    \[\leadsto u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \]
14. Add Preprocessing

Alternative 8: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u1 around inf 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. times-frac 98.3%

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

   3. sub-neg 98.3%

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

   4. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.4%

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

   2. lft-mult-inverse 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in u2 around 0 80.2%

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

   1. associate-*r* 80.2%

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

   2. sub-neg 80.2%

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

   3. metadata-eval 80.2%

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

   4. unpow-1 80.2%

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

   5. metadata-eval 80.2%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   6. pow-sqr 80.3%

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

   7. rem-sqrt-square 80.3%

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

   8. rem-square-sqrt 79.9%

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

   9. fabs-sqr 79.9%

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

   10. rem-square-sqrt 80.3%

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

10. Simplified 80.3%

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

11. Final simplification 80.3%

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

Alternative 9: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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 = u2 * ((((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * 6.28318530718e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u1 around inf 98.4%

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

   1. *-un-lft-identity 98.4%

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

   2. times-frac 98.3%

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

   3. sub-neg 98.3%

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

   4. metadata-eval 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.4%

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

   2. lft-mult-inverse 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in u2 around 0 88.5%

   \[\leadsto \color{blue}{u2 \cdot \left(-41.341702240407926 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 88.5%

      \[\leadsto u2 \cdot \left(\color{blue}{\left(-41.341702240407926 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + 6.28318530718 \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) \]

   2. distribute-rgt-out 88.5%

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

   3. sub-neg 88.5%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   4. metadata-eval 88.5%

      \[\leadsto u2 \cdot \left(\sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   5. unpow-1 88.5%

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-1}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   6. metadata-eval 88.5%

      \[\leadsto u2 \cdot \left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   7. pow-sqr 88.6%

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   8. rem-sqrt-square 88.6%

      \[\leadsto u2 \cdot \left(\color{blue}{\left|{\left(\frac{1}{u1} + -1\right)}^{-0.5}\right|} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   9. rem-square-sqrt 88.1%

      \[\leadsto u2 \cdot \left(\left|\color{blue}{\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}}\right| \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   10. fabs-sqr 88.1%

       \[\leadsto u2 \cdot \left(\color{blue}{\left(\sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \sqrt{{\left(\frac{1}{u1} + -1\right)}^{-0.5}}\right)} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

   11. rem-square-sqrt 88.6%

       \[\leadsto u2 \cdot \left(\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right) \]

10. Simplified 88.6%

    \[\leadsto \color{blue}{u2 \cdot \left({\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \left(-41.341702240407926 \cdot {u2}^{2} + 6.28318530718\right)\right)} \]

11. Taylor expanded in u2 around 0 80.3%

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

Alternative 10: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 80.0%

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

   1. *-commutative 80.0%

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

   2. associate-*r* 80.2%

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

5. Simplified 80.2%

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

Alternative 11: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 80.0%

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

4. Final simplification 80.0%

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

Alternative 12: 73.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt (* u1 (+ 1.0 u1))))))

C

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

Fortran

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 * Float32(Float32(1.0) + u1)))))
end

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 80.0%

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

4. Taylor expanded in u1 around 0 72.8%

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

   1. +-commutative 87.8%

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

6. Simplified 72.8%

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

7. Final simplification 72.8%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\right) \]
8. Add Preprocessing

Alternative 13: 65.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (* 6.28318530718 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (6.28318530718f * 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 = u2 * (6.28318530718e0 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(6.28318530718) * sqrt(u1)))
end

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 80.0%

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

4. Taylor expanded in u1 around 0 64.6%

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

   1. add-sqr-sqrt 64.5%

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

   2. sqrt-unprod 64.6%

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

   3. *-commutative 64.6%

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

   4. *-commutative 64.6%

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

   5. swap-sqr 64.6%

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

   6. swap-sqr 64.7%

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

   7. add-sqr-sqrt 64.7%

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

   8. pow2 64.7%

      \[\leadsto \sqrt{\left(u1 \cdot \color{blue}{{u2}^{2}}\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)} \]

   9. metadata-eval 64.7%

      \[\leadsto \sqrt{\left(u1 \cdot {u2}^{2}\right) \cdot \color{blue}{39.47841760436263}} \]

6. Applied egg-rr 64.7%

   \[\leadsto \color{blue}{\sqrt{\left(u1 \cdot {u2}^{2}\right) \cdot 39.47841760436263}} \]
7. Step-by-step derivation

   1. associate-*l* 64.7%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left({u2}^{2} \cdot 39.47841760436263\right)}} \]

8. Simplified 64.7%

   \[\leadsto \color{blue}{\sqrt{u1 \cdot \left({u2}^{2} \cdot 39.47841760436263\right)}} \]
9. Step-by-step derivation

   1. sqrt-prod 64.6%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sqrt{{u2}^{2} \cdot 39.47841760436263}} \]

   2. sqrt-prod 64.5%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\sqrt{{u2}^{2}} \cdot \sqrt{39.47841760436263}\right)} \]

   3. sqrt-pow1 64.5%

      \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{{u2}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{39.47841760436263}\right) \]

   4. metadata-eval 64.5%

      \[\leadsto \sqrt{u1} \cdot \left({u2}^{\color{blue}{1}} \cdot \sqrt{39.47841760436263}\right) \]

   5. pow1 64.5%

      \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{u2} \cdot \sqrt{39.47841760436263}\right) \]

   6. metadata-eval 64.7%

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

   7. *-commutative 64.7%

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

   8. associate-*r* 64.7%

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

   9. *-commutative 64.7%

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

10. Applied egg-rr 64.7%

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

11. Final simplification 64.7%

    \[\leadsto u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right) \]
12. Add Preprocessing

Alternative 14: 65.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 6.28318530718 u2) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * 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 = (6.28318530718e0 * u2) * sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * u2) * sqrt(u1))
end

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u1 around 0 76.5%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)} \]

4. Taylor expanded in u2 around 0 64.7%

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

5. Final simplification 64.7%

   \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1} \]
6. Add Preprocessing

Alternative 15: 65.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * 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 = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(u1)))
end

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 80.0%

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

4. Taylor expanded in u1 around 0 64.6%

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

5. Final simplification 64.6%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Add Preprocessing

Alternative 16: 4.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 = (u2 * sqrt(u1)) * (-6.28318530718e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0 80.0%

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

4. Taylor expanded in u1 around 0 64.6%

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

   1. add-sqr-sqrt 64.5%

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

   2. sqrt-unprod 64.6%

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

   3. *-commutative 64.6%

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

   4. *-commutative 64.6%

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

   5. swap-sqr 64.6%

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

   6. swap-sqr 64.7%

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

   7. add-sqr-sqrt 64.7%

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

   8. pow2 64.7%

      \[\leadsto \sqrt{\left(u1 \cdot \color{blue}{{u2}^{2}}\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)} \]

   9. metadata-eval 64.7%

      \[\leadsto \sqrt{\left(u1 \cdot {u2}^{2}\right) \cdot \color{blue}{39.47841760436263}} \]

6. Applied egg-rr 64.7%

   \[\leadsto \color{blue}{\sqrt{\left(u1 \cdot {u2}^{2}\right) \cdot 39.47841760436263}} \]
7. Step-by-step derivation

   1. associate-*l* 64.7%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left({u2}^{2} \cdot 39.47841760436263\right)}} \]

8. Simplified 64.7%

   \[\leadsto \color{blue}{\sqrt{u1 \cdot \left({u2}^{2} \cdot 39.47841760436263\right)}} \]

9. Taylor expanded in u2 around -inf 4.6%

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

10. Final simplification 4.6%

    \[\leadsto \left(u2 \cdot \sqrt{u1}\right) \cdot -6.28318530718 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024172 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :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))) (sin (* 6.28318530718 u2))))