Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 8.4s

Alternatives: 7

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: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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. frac-2neg N/A

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

   3. neg-sub0 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0 - u1}}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. div-sub N/A

      \[\leadsto \sqrt{\color{blue}{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. distribute-frac-neg2 N/A

      \[\leadsto \sqrt{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \color{blue}{\left(\mathsf{neg}\left(\frac{u1}{1 - u1}\right)\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. distribute-frac-neg N/A

      \[\leadsto \sqrt{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \color{blue}{\frac{\mathsf{neg}\left(u1\right)}{1 - u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. frac-sub N/A

      \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. lower-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right)} - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. neg-sub0 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(0 - \left(1 - u1\right)\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. lift--.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(1 - u1\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   14. sub-neg N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   15. +-commutative N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(u1\right)\right) + 1\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   16. associate--r+ N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(u1\right)\right)\right) - 1\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   17. neg-sub0 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(u1\right)\right)\right)\right)} - 1\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   18. remove-double-neg N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(\color{blue}{u1} - 1\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   19. lower--.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(u1 - 1\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   20. lower-neg.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \color{blue}{\left(-u1\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   21. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \left(-u1\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

4. Applied rewrites 99.1%

   \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \left(-u1\right)}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
5. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \left(-u1\right)}}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right)} - \left(u1 - 1\right) \cdot \left(-u1\right)}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. mul0-lft N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0} - \left(u1 - 1\right) \cdot \left(-u1\right)}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. neg-sub0 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. remove-double-neg N/A

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

   9. lower-*.f32 99.1

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

6. Applied rewrites 99.1%

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

Alternative 2: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999924898147583:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999924898147583)
     (* (sqrt u1) t_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 <= 0.9999924898147583f) {
		tmp = sqrtf(u1) * t_0;
	} else {
		tmp = 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 <= 0.9999924898147583e0) then
        tmp = sqrt(u1) * t_0
    else
        tmp = 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(0.9999924898147583))
		tmp = Float32(sqrt(u1) * t_0);
	else
		tmp = 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(0.9999924898147583))
		tmp = sqrt(u1) * t_0;
	else
		tmp = 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)) < 0.99999249

   1. Initial program 98.1%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   if 0.99999249 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   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 97.7%

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

Alternative 3: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0014949999749660492:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(u1 - -1\right) \cdot u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0014949999749660492)
   (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) <= 0.0014949999749660492f) {
		tmp = 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) <= 0.0014949999749660492e0) then
        tmp = 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(0.0014949999749660492))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(sqrt(Float32(Float32(u1 - 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(0.0014949999749660492))
		tmp = 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) < 0.001495

   1. Initial program 99.5%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   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 98.9%

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

   if 0.001495 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 - u1}}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. div-sub N/A

         \[\leadsto \sqrt{\color{blue}{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. distribute-frac-neg2 N/A

         \[\leadsto \sqrt{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \color{blue}{\left(\mathsf{neg}\left(\frac{u1}{1 - u1}\right)\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \sqrt{\frac{0}{\mathsf{neg}\left(\left(1 - u1\right)\right)} - \color{blue}{\frac{\mathsf{neg}\left(u1\right)}{1 - u1}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      7. frac-sub N/A

         \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      9. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right) - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right)} - \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      12. neg-sub0 N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(0 - \left(1 - u1\right)\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      13. lift--.f32 N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(1 - u1\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      14. sub-neg N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      15. +-commutative N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(u1\right)\right) + 1\right)}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      16. associate--r+ N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(u1\right)\right)\right) - 1\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      17. neg-sub0 N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(u1\right)\right)\right)\right)} - 1\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      18. remove-double-neg N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(\color{blue}{u1} - 1\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      19. lower--.f32 N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \color{blue}{\left(u1 - 1\right)} \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      20. lower-neg.f32 N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \color{blue}{\left(-u1\right)}}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      21. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \left(-u1\right)}{\color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(1 - u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \left(-u1\right)}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right) - \left(u1 - 1\right) \cdot \left(-u1\right)}}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 \cdot \left(1 - u1\right)} - \left(u1 - 1\right) \cdot \left(-u1\right)}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. mul0-lft N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0} - \left(u1 - 1\right) \cdot \left(-u1\right)}{\left(u1 - 1\right) \cdot \left(1 - u1\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. neg-sub0 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. remove-double-neg N/A

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

      9. lower-*.f32 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. times-frac N/A

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

      5. *-inverses N/A

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

      6. associate-*r/ N/A

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

      7. *-lft-identity N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 97.9

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in u1 around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. sub-neg N/A

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

       4. lower--.f32 88.0

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

   11. Applied rewrites 88.0%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

Alternative 5: 79.9% 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 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   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 78.2%

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

Alternative 6: 71.5% accurate, 7.1× speedup

Math

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

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(Float32(u1 - 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 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   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 78.2%

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

7. Applied rewrites 78.1%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 70.1%

    \[\leadsto \sqrt{\left(u1 - -1\right) \cdot u1} \]
11. Add Preprocessing

Alternative 7: 63.1% 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 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   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 78.2%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 62.8%

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

Reproduce

herbie shell --seed 2024303 
(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))))