Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 8.4s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{\left(1 - u1\right) \cdot \left(1 - u1\right)}{u1 \cdot u1}\right)}^{-0.25} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (cos (* 6.28318530718 u2))
  (pow (/ (* (- 1.0 u1) (- 1.0 u1)) (* u1 u1)) -0.25)))

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((6.28318530718f * u2)) * powf((((1.0f - u1) * (1.0f - u1)) / (u1 * u1)), -0.25f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = cos((6.28318530718e0 * u2)) * ((((1.0e0 - u1) * (1.0e0 - u1)) / (u1 * u1)) ** (-0.25e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cos(Float32(Float32(6.28318530718) * u2)) * (Float32(Float32(Float32(Float32(1.0) - u1) * Float32(Float32(1.0) - u1)) / Float32(u1 * u1)) ^ Float32(-0.25)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = cos((single(6.28318530718) * u2)) * ((((single(1.0) - u1) * (single(1.0) - u1)) / (u1 * u1)) ^ single(-0.25));
end

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{{\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}^{-0.25}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto {\color{blue}{\left(\frac{u1 - 1}{u1} \cdot \frac{u1 - 1}{u1}\right)}}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. lift-/.f32 N/A

      \[\leadsto {\left(\color{blue}{\frac{u1 - 1}{u1}} \cdot \frac{u1 - 1}{u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. frac-2neg N/A

      \[\leadsto {\left(\color{blue}{\frac{\mathsf{neg}\left(\left(u1 - 1\right)\right)}{\mathsf{neg}\left(u1\right)}} \cdot \frac{u1 - 1}{u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. lift-neg.f32 N/A

      \[\leadsto {\left(\frac{\mathsf{neg}\left(\left(u1 - 1\right)\right)}{\color{blue}{-u1}} \cdot \frac{u1 - 1}{u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. lift-/.f32 N/A

      \[\leadsto {\left(\frac{\mathsf{neg}\left(\left(u1 - 1\right)\right)}{-u1} \cdot \color{blue}{\frac{u1 - 1}{u1}}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. frac-times N/A

      \[\leadsto {\color{blue}{\left(\frac{\left(\mathsf{neg}\left(\left(u1 - 1\right)\right)\right) \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. lift--.f32 N/A

      \[\leadsto {\left(\frac{\left(\mathsf{neg}\left(\color{blue}{\left(u1 - 1\right)}\right)\right) \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. sub-neg N/A

      \[\leadsto {\left(\frac{\left(\mathsf{neg}\left(\color{blue}{\left(u1 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. metadata-eval N/A

      \[\leadsto {\left(\frac{\left(\mathsf{neg}\left(\left(u1 + \color{blue}{-1}\right)\right)\right) \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. distribute-neg-in N/A

       \[\leadsto {\left(\frac{\color{blue}{\left(\left(\mathsf{neg}\left(u1\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. lift-neg.f32 N/A

       \[\leadsto {\left(\frac{\left(\color{blue}{\left(-u1\right)} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. metadata-eval N/A

       \[\leadsto {\left(\frac{\left(\left(-u1\right) + \color{blue}{1}\right) \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. +-commutative N/A

       \[\leadsto {\left(\frac{\color{blue}{\left(1 + \left(-u1\right)\right)} \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   14. lift-neg.f32 N/A

       \[\leadsto {\left(\frac{\left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right) \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. sub-neg N/A

       \[\leadsto {\left(\frac{\color{blue}{\left(1 - u1\right)} \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   16. lift--.f32 N/A

       \[\leadsto {\left(\frac{\color{blue}{\left(1 - u1\right)} \cdot \left(u1 - 1\right)}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   17. *-commutative N/A

       \[\leadsto {\left(\frac{\color{blue}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   18. lift-*.f32 N/A

       \[\leadsto {\left(\frac{\color{blue}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}}{\left(-u1\right) \cdot u1}\right)}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   19. lower-/.f32 N/A

       \[\leadsto {\color{blue}{\left(\frac{\left(u1 - 1\right) \cdot \left(1 - u1\right)}{\left(-u1\right) \cdot u1}\right)}}^{\frac{-1}{4}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   20. lower-*.f32 99.2

       \[\leadsto {\left(\frac{\left(u1 - 1\right) \cdot \left(1 - u1\right)}{\color{blue}{\left(-u1\right) \cdot u1}}\right)}^{-0.25} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

6. Applied rewrites 99.2%

   \[\leadsto {\color{blue}{\left(\frac{\left(u1 - 1\right) \cdot \left(1 - u1\right)}{\left(-u1\right) \cdot u1}\right)}}^{-0.25} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.2%

   \[\leadsto \color{blue}{{\left(\frac{\left(1 - u1\right) \cdot \left(1 - u1\right)}{u1 \cdot u1}\right)}^{-0.25} \cdot \cos \left(u2 \cdot 6.28318530718\right)} \]

9. Final simplification 99.2%

   \[\leadsto \cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{\left(1 - u1\right) \cdot \left(1 - u1\right)}{u1 \cdot u1}\right)}^{-0.25} \]
10. Add Preprocessing

Alternative 2: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9977999925613403:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317 + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9977999925613403)
     (* (sqrt u1) t_0)
     (* (+ (* (* u2 u2) -19.739208802181317) 1.0) (sqrt (/ u1 (- 1.0 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9977999925613403f) {
		tmp = sqrtf(u1) * t_0;
	} else {
		tmp = (((u2 * u2) * -19.739208802181317f) + 1.0f) * 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) :: t_0
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9977999925613403e0) then
        tmp = sqrt(u1) * t_0
    else
        tmp = (((u2 * u2) * (-19.739208802181317e0)) + 1.0e0) * sqrt((u1 / (1.0e0 - u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9977999925613403))
		tmp = Float32(sqrt(u1) * t_0);
	else
		tmp = Float32(Float32(Float32(Float32(u2 * u2) * Float32(-19.739208802181317)) + Float32(1.0)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9977999925613403))
		tmp = sqrt(u1) * t_0;
	else
		tmp = (((u2 * u2) * single(-19.739208802181317)) + single(1.0)) * sqrt((u1 / (single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.997799993

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.9

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

   5. Applied rewrites 76.9%

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

   if 0.997799993 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. *-rgt-identity N/A

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

      10. sub-neg N/A

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

      11. rgt-mult-inverse N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. mul-1-neg N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-lft-in N/A

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. associate-*r* N/A

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

   5. Applied rewrites 91.2%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9977999925613403:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317 + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.05000000074505806:\\ \;\;\;\;\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317 + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-u1\right) \cdot \left(-1 - u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.05000000074505806)
   (* (+ (* (* u2 u2) -19.739208802181317) 1.0) (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt (* (- u1) (- -1.0 u1))) (cos (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.05000000074505806f) {
		tmp = (((u2 * u2) * -19.739208802181317f) + 1.0f) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf((-u1 * (-1.0f - u1))) * cosf((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) <= 0.05000000074505806e0) then
        tmp = (((u2 * u2) * (-19.739208802181317e0)) + 1.0e0) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt((-u1 * ((-1.0e0) - u1))) * cos((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(0.05000000074505806))
		tmp = Float32(Float32(Float32(Float32(u2 * u2) * Float32(-19.739208802181317)) + Float32(1.0)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(Float32(Float32(-u1) * Float32(Float32(-1.0) - u1))) * cos(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(0.05000000074505806))
		tmp = (((u2 * u2) * single(-19.739208802181317)) + single(1.0)) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt((-u1 * (single(-1.0) - u1))) * cos((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) < 0.0500000007

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. *-rgt-identity N/A

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

      10. sub-neg N/A

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

      11. rgt-mult-inverse N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. mul-1-neg N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-lft-in N/A

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. associate-*r* N/A

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

   5. Applied rewrites 92.2%

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

   7. Applied rewrites 99.2%

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

   if 0.0500000007 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 N/A

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

      9. lower-neg.f32 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in u1 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 87.2

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

   7. Applied rewrites 87.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.05000000074505806:\\ \;\;\;\;\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317 + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-u1\right) \cdot \left(-1 - u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 5: 88.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. *-rgt-identity N/A

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

   10. sub-neg N/A

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

   11. rgt-mult-inverse N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. *-rgt-identity N/A

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

   16. distribute-lft-in N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. associate-*r* N/A

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

5. Applied rewrites 79.3%

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

7. Applied rewrites 88.6%

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

8. Final simplification 88.6%

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

Alternative 6: 80.2% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 79.7%

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

Alternative 7: 71.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 79.7%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 61.5%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 69.3%

    \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right) \cdot u1 + u1} \]
11. Add Preprocessing

Alternative 8: 63.4% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 79.7%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 61.5%

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

Reproduce

herbie shell --seed 2024282 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))