Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 9.4s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

4. Applied rewrites 98.1%

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

   1. lift-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{-1}{\frac{u1 - 1}{u1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. lift-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{-1}{\frac{u1 - 1}{u1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. frac-2neg N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{u1 - 1}{u1}\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\color{blue}{1}}{\mathsf{neg}\left(\frac{u1 - 1}{u1}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. sqrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{neg}\left(\frac{u1 - 1}{u1}\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{\mathsf{neg}\left(\frac{u1 - 1}{u1}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\mathsf{neg}\left(\frac{u1 - 1}{u1}\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. lower-sqrt.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\mathsf{neg}\left(\frac{u1 - 1}{u1}\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\mathsf{neg}\left(\color{blue}{\frac{u1 - 1}{u1}}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. distribute-neg-frac N/A

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(u1 - 1\right)\right)}{u1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. lift--.f32 N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(u1 - 1\right)}\right)}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. sub-neg N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(u1 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. metadata-eval N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\mathsf{neg}\left(\left(u1 + \color{blue}{-1}\right)\right)}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)}}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. metadata-eval N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(\mathsf{neg}\left(u1\right)\right) + \color{blue}{1}}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   16. +-commutative N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   17. sub-neg N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{1 - u1}}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   18. lift--.f32 N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{1 - u1}}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   19. lower-/.f32 98.1

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

6. Applied rewrites 98.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{1}{\sqrt{\frac{1 - u1}{u1}}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \]

   4. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \cdot \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \]

   5. lift-*.f32 N/A

      \[\leadsto \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \cdot \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \]

   6. lift-/.f32 N/A

      \[\leadsto \sin \left(u2 \cdot \frac{314159265359}{50000000000}\right) \cdot \color{blue}{\frac{1}{\sqrt{\frac{1 - u1}{u1}}}} \]

   7. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   8. lower-/.f32 98.2

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

8. Applied rewrites 98.2%

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

9. Final simplification 98.2%

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

Alternative 2: 95.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\left(u1 + 1\right) \cdot u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.002199999988079071)
   (* (sqrt (* (+ u1 1.0) u1)) (sin (* 6.28318530718 u2)))
   (/
    (* (+ (* -41.341702240407926 (* u2 u2)) 6.28318530718) u2)
    (sqrt (/ (- 1.0 u1) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.002199999988079071f) {
		tmp = sqrtf(((u1 + 1.0f) * u1)) * sinf((6.28318530718f * u2));
	} else {
		tmp = (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * u2) / sqrtf(((1.0f - u1) / 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 ((u1 / (1.0e0 - u1)) <= 0.002199999988079071e0) then
        tmp = sqrt(((u1 + 1.0e0) * u1)) * sin((6.28318530718e0 * u2))
    else
        tmp = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2) / sqrt(((1.0e0 - u1) / u1))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.002199999988079071))
		tmp = Float32(sqrt(Float32(Float32(u1 + Float32(1.0)) * u1)) * sin(Float32(Float32(6.28318530718) * u2)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) + Float32(6.28318530718)) * u2) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u1 / (single(1.0) - u1)) <= single(0.002199999988079071))
		tmp = sqrt(((u1 + single(1.0)) * u1)) * sin((single(6.28318530718) * u2));
	else
		tmp = (((single(-41.341702240407926) * (u2 * u2)) + single(6.28318530718)) * u2) / sqrt(((single(1.0) - u1) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.0022

   1. Initial program 97.9%

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

   4. Applied rewrites 98.0%

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

      1. lift-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{-1}{\frac{u1 - 1}{u1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. lift-/.f32 N/A

         \[\leadsto \sqrt{\frac{-1}{\color{blue}{\frac{u1 - 1}{u1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. associate-/r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{-1}{u1 - 1} \cdot u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. frac-2neg N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(u1 - 1\right)\right)}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\mathsf{neg}\left(\left(u1 - 1\right)\right)} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      6. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{neg}\left(\color{blue}{\left(u1 - 1\right)}\right)} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      7. sub-neg N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{neg}\left(\color{blue}{\left(u1 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{neg}\left(\left(u1 + \color{blue}{-1}\right)\right)} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{\left(\mathsf{neg}\left(u1\right)\right) + \color{blue}{1}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      12. sub-neg N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - u1}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      13. lift--.f32 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{1 - u1}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{1 - u1} \cdot u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      15. lower-/.f32 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1\right)} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   8. Step-by-step derivation

      1. lower-+.f32 97.6

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

   9. Applied rewrites 97.6%

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

   if 0.0022 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.4%

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

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. lift-sqrt.f32 N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

      4. lift-/.f32 N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

      5. sqrt-div N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{u1}}{\sqrt{1 - u1}}} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

      10. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{1 - u1}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

      14. lower-sqrt.f32 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}}{\sqrt{1 - u1}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\left(\color{blue}{{u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}} + \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

      7. lower-*.f32 83.0

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

   7. Applied rewrites 82.4%

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \sqrt{u1}}}{\sqrt{1 - u1}} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{1 - u1}} \cdot \sqrt{u1}} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{1 - u1}}\right)} \cdot \sqrt{u1} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \left(\frac{1}{\sqrt{1 - u1}} \cdot \sqrt{u1}\right)} \]

      7. associate-/r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}} \]

      8. lift-sqrt.f32 N/A

         \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 - u1}}}{\sqrt{u1}}} \]

      9. lift-sqrt.f32 N/A

         \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\sqrt{1 - u1}}{\color{blue}{\sqrt{u1}}}} \]

      10. sqrt-div N/A

          \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

      11. lift-/.f32 N/A

          \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{1 - u1}{u1}}}} \]

      12. lift-sqrt.f32 N/A

          \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

      13. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

      14. lower-/.f32 83.4

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

   9. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 91.7%

       \[\leadsto \frac{\left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 + 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\left(u1 + 1\right) \cdot u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 98.1%

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

3. Final simplification 98.1%

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

Alternative 4: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.36000001430511475:\\ \;\;\;\;\frac{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2 + 6.28318530718 \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.36000001430511475)
   (/
    (+ (* (* -41.341702240407926 (* u2 u2)) u2) (* 6.28318530718 u2))
    (sqrt (/ (- 1.0 u1) u1)))
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.36000001430511475f) {
		tmp = (((-41.341702240407926f * (u2 * u2)) * u2) + (6.28318530718f * u2)) / sqrtf(((1.0f - u1) / u1));
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	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) <= 0.36000001430511475e0) then
        tmp = ((((-41.341702240407926e0) * (u2 * u2)) * u2) + (6.28318530718e0 * u2)) / sqrt(((1.0e0 - u1) / u1))
    else
        tmp = sqrt(u1) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.36000001430511475))
		tmp = Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) * u2) + Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.36000001430511475))
		tmp = (((single(-41.341702240407926) * (u2 * u2)) * u2) + (single(6.28318530718) * u2)) / sqrt(((single(1.0) - u1) / u1));
	else
		tmp = sqrt(u1) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.360000014

   1. Initial program 98.4%

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

      1. lift-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. lift-sqrt.f32 N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

      4. lift-/.f32 N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

      5. sqrt-div N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{u1}}{\sqrt{1 - u1}}} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

      7. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

      10. lower-sqrt.f32 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{1 - u1}} \]

      11. lift-*.f32 N/A

          \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

      14. lower-sqrt.f32 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}}{\sqrt{1 - u1}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\left(\color{blue}{{u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}} + \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

      5. lower-fma.f32 N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

      7. lower-*.f32 87.8

         \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

   7. Applied rewrites 87.6%

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}}} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \sqrt{u1}}}{\sqrt{1 - u1}} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{1 - u1}} \cdot \sqrt{u1}} \]

      5. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{1 - u1}}\right)} \cdot \sqrt{u1} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \left(\frac{1}{\sqrt{1 - u1}} \cdot \sqrt{u1}\right)} \]

      7. associate-/r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}} \]

      8. lift-sqrt.f32 N/A

         \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 - u1}}}{\sqrt{u1}}} \]

      9. lift-sqrt.f32 N/A

         \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\sqrt{1 - u1}}{\color{blue}{\sqrt{u1}}}} \]

      10. sqrt-div N/A

          \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

      11. lift-/.f32 N/A

          \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{1 - u1}{u1}}}} \]

      12. lift-sqrt.f32 N/A

          \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

      13. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

      14. lower-/.f32 88.2

          \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

   9. Applied rewrites 88.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.4%

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

   if 0.360000014 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.4%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.36000001430511475:\\ \;\;\;\;\frac{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2 + 6.28318530718 \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (((-41.341702240407926f * (u2 * u2)) * u2) + (6.28318530718f * u2)) / sqrtf(((1.0f - u1) / 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 = ((((-41.341702240407926e0) * (u2 * u2)) * u2) + (6.28318530718e0 * u2)) / sqrt(((1.0e0 - u1) / u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. lift-sqrt.f32 N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

   5. sqrt-div N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{u1}}{\sqrt{1 - u1}}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   10. lower-sqrt.f32 N/A

       \[\leadsto \frac{\color{blue}{\sqrt{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{1 - u1}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

   14. lower-sqrt.f32 97.8

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}}{\sqrt{1 - u1}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\left(\color{blue}{{u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}} + \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

   7. lower-*.f32 76.7

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

7. Applied rewrites 78.2%

   \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \sqrt{u1}}}{\sqrt{1 - u1}} \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{1 - u1}} \cdot \sqrt{u1}} \]

   5. div-inv N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{1 - u1}}\right)} \cdot \sqrt{u1} \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \left(\frac{1}{\sqrt{1 - u1}} \cdot \sqrt{u1}\right)} \]

   7. associate-/r/ N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}} \]

   8. lift-sqrt.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 - u1}}}{\sqrt{u1}}} \]

   9. lift-sqrt.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\sqrt{1 - u1}}{\color{blue}{\sqrt{u1}}}} \]

   10. sqrt-div N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   11. lift-/.f32 N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{1 - u1}{u1}}}} \]

   12. lift-sqrt.f32 N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   13. un-div-inv N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

   14. lower-/.f32 78.7

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

9. Applied rewrites 78.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]
10. Step-by-step derivation

11. Applied rewrites 86.7%

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

12. Final simplification 86.7%

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

Alternative 6: 88.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * u2) / sqrtf(((1.0f - u1) / 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 = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2) / sqrt(((1.0e0 - u1) / u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. lift-sqrt.f32 N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

   5. sqrt-div N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{u1}}{\sqrt{1 - u1}}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   10. lower-sqrt.f32 N/A

       \[\leadsto \frac{\color{blue}{\sqrt{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{1 - u1}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

   14. lower-sqrt.f32 97.8

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}}{\sqrt{1 - u1}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\left(\color{blue}{{u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}} + \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

   7. lower-*.f32 78.4

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

7. Applied rewrites 78.2%

   \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \sqrt{u1}}}{\sqrt{1 - u1}} \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{1 - u1}} \cdot \sqrt{u1}} \]

   5. div-inv N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{1 - u1}}\right)} \cdot \sqrt{u1} \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \left(\frac{1}{\sqrt{1 - u1}} \cdot \sqrt{u1}\right)} \]

   7. associate-/r/ N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}} \]

   8. lift-sqrt.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 - u1}}}{\sqrt{u1}}} \]

   9. lift-sqrt.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\sqrt{1 - u1}}{\color{blue}{\sqrt{u1}}}} \]

   10. sqrt-div N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   11. lift-/.f32 N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{1 - u1}{u1}}}} \]

   12. lift-sqrt.f32 N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   13. un-div-inv N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

   14. lower-/.f32 78.7

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

9. Applied rewrites 78.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]
10. Step-by-step derivation

11. Applied rewrites 86.7%

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

12. Final simplification 86.7%

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

Alternative 7: 81.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. lift-sqrt.f32 N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

   5. sqrt-div N/A

      \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{u1}}{\sqrt{1 - u1}}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

   7. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   10. lower-sqrt.f32 N/A

       \[\leadsto \frac{\color{blue}{\sqrt{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{1 - u1}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}{\sqrt{1 - u1}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{\sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)}}{\sqrt{1 - u1}} \]

   14. lower-sqrt.f32 97.8

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}}{\sqrt{1 - u1}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\left(\color{blue}{{u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}} + \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

   5. lower-fma.f32 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)} \cdot u2\right)}{\sqrt{1 - u1}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

   7. lower-*.f32 78.4

      \[\leadsto \frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}{\sqrt{1 - u1}} \]

7. Applied rewrites 78.4%

   \[\leadsto \frac{\sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}{\sqrt{1 - u1}}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{u1} \cdot \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right)}}{\sqrt{1 - u1}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \sqrt{u1}}}{\sqrt{1 - u1}} \]

   4. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{1 - u1}} \cdot \sqrt{u1}} \]

   5. div-inv N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{1 - u1}}\right)} \cdot \sqrt{u1} \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \left(\frac{1}{\sqrt{1 - u1}} \cdot \sqrt{u1}\right)} \]

   7. associate-/r/ N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}} \]

   8. lift-sqrt.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 - u1}}}{\sqrt{u1}}} \]

   9. lift-sqrt.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\frac{\sqrt{1 - u1}}{\color{blue}{\sqrt{u1}}}} \]

   10. sqrt-div N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   11. lift-/.f32 N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\sqrt{\color{blue}{\frac{1 - u1}{u1}}}} \]

   12. lift-sqrt.f32 N/A

       \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \cdot \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

   13. un-div-inv N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

   14. lower-/.f32 78.7

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

9. Applied rewrites 78.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}} \]

10. Taylor expanded in u2 around 0

    \[\leadsto \frac{\frac{314159265359}{50000000000} \cdot u2}{\sqrt{\frac{1 - u1}{u1}}} \]
11. Step-by-step derivation

12. Applied rewrites 78.7%

    \[\leadsto \frac{6.28318530718 \cdot u2}{\sqrt{\frac{1 - u1}{u1}}} \]
13. Add Preprocessing

Alternative 8: 81.1% accurate, 3.9× 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 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 78.6

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

5. Applied rewrites 78.6%

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

6. Final simplification 78.6%

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

Alternative 9: 64.2% accurate, 6.4× 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 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 78.6

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

5. Applied rewrites 78.6%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 62.3

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

8. Applied rewrites 62.3%

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

9. Final simplification 62.3%

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

Reproduce

herbie shell --seed 2024276 
(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))))