Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 8.6s

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: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = cos((u2 * single(6.28318530718))) * sqrt((u1 / (((u1 * u1) - single(1.0)) / (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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. sqr-neg N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-neg.f32 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((single(-1.0) / ((u1 - single(1.0)) / u1))) * cos((u2 * single(6.28318530718)));
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 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.1860000044107437:\\ \;\;\;\;\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317 + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-1 - u1\right) \cdot \left(-u1\right)} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.1860000044107437)
   (* (+ (* (* u2 u2) -19.739208802181317) 1.0) (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt (* (- -1.0 u1) (- u1))) (cos (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.1860000044107437f) {
		tmp = (((u2 * u2) * -19.739208802181317f) + 1.0f) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(((-1.0f - u1) * -u1)) * cosf((u2 * 6.28318530718f));
	}
	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 ((u2 * 6.28318530718e0) <= 0.1860000044107437e0) then
        tmp = (((u2 * u2) * (-19.739208802181317e0)) + 1.0e0) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt((((-1.0e0) - u1) * -u1)) * cos((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.1860000044107437))
		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(Float32(-1.0) - u1) * Float32(-u1))) * cos(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.1860000044107437))
		tmp = (((u2 * u2) * single(-19.739208802181317)) + single(1.0)) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(((single(-1.0) - u1) * -u1)) * cos((u2 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.186000004

   1. Initial program 99.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. div-inv N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. lower-pow.f32 N/A

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

      19. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 90.6%

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

   9. Applied rewrites 98.6%

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

   if 0.186000004 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 97.2

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

   4. Applied rewrites 97.2%

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

   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 84.9

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

   7. Applied rewrites 84.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.1860000044107437:\\ \;\;\;\;\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317 + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-1 - u1\right) \cdot \left(-u1\right)} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

Alternative 5: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.30000001192092896:\\ \;\;\;\;\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317 + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.30000001192092896)
   (* (+ (* (* u2 u2) -19.739208802181317) 1.0) (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt u1) (cos (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.30000001192092896f) {
		tmp = (((u2 * u2) * -19.739208802181317f) + 1.0f) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * cosf((u2 * 6.28318530718f));
	}
	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 ((u2 * 6.28318530718e0) <= 0.30000001192092896e0) then
        tmp = (((u2 * u2) * (-19.739208802181317e0)) + 1.0e0) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(u1) * cos((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.30000001192092896))
		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(u1) * cos(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.30000001192092896))
		tmp = (((u2 * u2) * single(-19.739208802181317)) + single(1.0)) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(u1) * cos((u2 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.300000012

   1. Initial program 99.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. div-inv N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. lower-pow.f32 N/A

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

      19. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 98.1%

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

   if 0.300000012 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.30000001192092896:\\ \;\;\;\;\left(\left(u2 \cdot u2\right) \cdot -19.739208802181317 + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. sqrt-prod N/A

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

   7. pow1/2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. inv-pow N/A

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

   17. pow-pow N/A

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

   18. lower-pow.f32 N/A

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

   19. metadata-eval 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

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

7. Applied rewrites 80.6%

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

9. Applied rewrites 88.8%

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

10. Final simplification 88.8%

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

Alternative 7: 79.8% 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 81.0%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
6. 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 81.0%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 64.1%

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

Reproduce

herbie shell --seed 2024243 
(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))))