Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 14.6s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

4. Applied rewrites 98.7%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. sqr-neg N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-neg.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.4000000059604645:\\ \;\;\;\;\sqrt{\frac{\frac{1}{u1 + -1}}{\frac{-1}{u1}}} \cdot \left(u2 \cdot \left(t\_0 \cdot \left(\frac{6.28318530718}{t\_0} + \left(81.6052492761019 + \frac{-41.341702240407926}{u2 \cdot u2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* u2 u2) (* u2 u2))))
   (if (<= (* 6.28318530718 u2) 0.4000000059604645)
     (*
      (sqrt (/ (/ 1.0 (+ u1 -1.0)) (/ -1.0 u1)))
      (*
       u2
       (*
        t_0
        (+
         (/ 6.28318530718 t_0)
         (+ 81.6052492761019 (/ -41.341702240407926 (* u2 u2)))))))
     (* (sin (* 6.28318530718 u2)) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (u2 * u2) * (u2 * u2);
	float tmp;
	if ((6.28318530718f * u2) <= 0.4000000059604645f) {
		tmp = sqrtf(((1.0f / (u1 + -1.0f)) / (-1.0f / u1))) * (u2 * (t_0 * ((6.28318530718f / t_0) + (81.6052492761019f + (-41.341702240407926f / (u2 * u2))))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (u2 * u2) * (u2 * u2)
    if ((6.28318530718e0 * u2) <= 0.4000000059604645e0) then
        tmp = sqrt(((1.0e0 / (u1 + (-1.0e0))) / ((-1.0e0) / u1))) * (u2 * (t_0 * ((6.28318530718e0 / t_0) + (81.6052492761019e0 + ((-41.341702240407926e0) / (u2 * u2))))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(u2 * u2) * Float32(u2 * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.4000000059604645))
		tmp = Float32(sqrt(Float32(Float32(Float32(1.0) / Float32(u1 + Float32(-1.0))) / Float32(Float32(-1.0) / u1))) * Float32(u2 * Float32(t_0 * Float32(Float32(Float32(6.28318530718) / t_0) + Float32(Float32(81.6052492761019) + Float32(Float32(-41.341702240407926) / Float32(u2 * u2)))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = (u2 * u2) * (u2 * u2);
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.4000000059604645))
		tmp = sqrt(((single(1.0) / (u1 + single(-1.0))) / (single(-1.0) / u1))) * (u2 * (t_0 * ((single(6.28318530718) / t_0) + (single(81.6052492761019) + (single(-41.341702240407926) / (u2 * u2))))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.400000006

   1. Initial program 98.8%

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

      1. lift--.f32 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f32 N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-+.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-/.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. lower-*.f32 88.3

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]

   8. Taylor expanded in u2 around inf

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{4} \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{4} \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left({u2}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left({u2}^{2} \cdot {u2}^{2}\right)} \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left({u2}^{2} \cdot {u2}^{2}\right)} \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot {u2}^{2}\right) \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot {u2}^{2}\right) \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \color{blue}{\left(u2 \cdot u2\right)}\right) \cdot \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{\frac{314159265359}{50000000000}}{{u2}^{4}}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\color{blue}{\left(\frac{\frac{314159265359}{50000000000}}{{u2}^{4}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right) \]

      10. associate--l+ N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \color{blue}{\left(\frac{\frac{314159265359}{50000000000}}{{u2}^{4}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)}\right)\right) \]

      11. lower-+.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \color{blue}{\left(\frac{\frac{314159265359}{50000000000}}{{u2}^{4}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)}\right)\right) \]

      12. lower-/.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\color{blue}{\frac{\frac{314159265359}{50000000000}}{{u2}^{4}}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{{u2}^{\color{blue}{\left(2 \cdot 2\right)}}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)\right) \]

      14. pow-sqr N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\color{blue}{{u2}^{2} \cdot {u2}^{2}}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)\right) \]

      15. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\color{blue}{{u2}^{2} \cdot {u2}^{2}}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\color{blue}{\left(u2 \cdot u2\right)} \cdot {u2}^{2}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)\right) \]

      17. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\color{blue}{\left(u2 \cdot u2\right)} \cdot {u2}^{2}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\left(u2 \cdot u2\right) \cdot \color{blue}{\left(u2 \cdot u2\right)}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)\right) \]

      19. lower-*.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\left(u2 \cdot u2\right) \cdot \color{blue}{\left(u2 \cdot u2\right)}} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)\right) \]

      20. sub-neg N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)} + \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)}\right)\right)\right) \]

      21. lower-+.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)} + \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \frac{1}{{u2}^{2}}\right)\right)\right)}\right)\right)\right) \]

      22. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot 1}{{u2}^{2}}}\right)\right)\right)\right)\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}}}{{u2}^{2}}\right)\right)\right)\right)\right)\right) \]

      24. distribute-neg-frac N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{\frac{314159265359}{50000000000}}{\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)} + \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}{{u2}^{2}}}\right)\right)\right)\right) \]

   10. Applied rewrites 97.3%

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \color{blue}{\left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{6.28318530718}{\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)} + \left(81.6052492761019 + \frac{-41.341702240407926}{u2 \cdot u2}\right)\right)\right)}\right) \]

   if 0.400000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. lower-*.f32 75.8

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

   5. Applied rewrites 75.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.4000000059604645:\\ \;\;\;\;\sqrt{\frac{\frac{1}{u1 + -1}}{\frac{-1}{u1}}} \cdot \left(u2 \cdot \left(\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\frac{6.28318530718}{\left(u2 \cdot u2\right) \cdot \left(u2 \cdot u2\right)} + \left(81.6052492761019 + \frac{-41.341702240407926}{u2 \cdot u2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Final simplification 98.7%

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

Alternative 5: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.1899999976158142:\\ \;\;\;\;t\_0 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right) + u2 \cdot \left(6.28318530718 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.1899999976158142)
     (+
      (*
       t_0
       (*
        (* u2 u2)
        (* u2 (fma u2 (* u2 81.6052492761019) -41.341702240407926))))
      (* u2 (* 6.28318530718 t_0)))
     (* (sin (* 6.28318530718 u2)) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 0.1899999976158142f) {
		tmp = (t_0 * ((u2 * u2) * (u2 * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f)))) + (u2 * (6.28318530718f * t_0));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.1899999976158142))
		tmp = Float32(Float32(t_0 * Float32(Float32(u2 * u2) * Float32(u2 * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))))) + Float32(u2 * Float32(Float32(6.28318530718) * t_0)));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.189999998

   1. Initial program 98.8%

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

      1. lift--.f32 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f32 N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-+.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-/.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. lower-*.f32 90.3

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 90.3%

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]

   9. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} + u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   if 0.189999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. lower-*.f32 76.3

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

   5. Applied rewrites 76.3%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.1899999976158142:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right) + u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ u2 \cdot \left(6.28318530718 \cdot t\_0\right) + \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right) \cdot \left(u2 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (+
    (* u2 (* 6.28318530718 t_0))
    (*
     (* (* u2 u2) (fma u2 (* u2 81.6052492761019) -41.341702240407926))
     (* u2 t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return (u2 * (6.28318530718f * t_0)) + (((u2 * u2) * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f)) * (u2 * t_0));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(Float32(u2 * Float32(Float32(6.28318530718) * t_0)) + Float32(Float32(Float32(u2 * u2) * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))) * Float32(u2 * t_0)))
end

Derivation

1. Initial program 98.7%

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

   1. lift--.f32 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. neg-sub0 N/A

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

   10. lift--.f32 N/A

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

   11. associate--r- N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. lower-+.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-/.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   12. lower-*.f32 78.0

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

7. Applied rewrites 78.0%

   \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]

9. Applied rewrites 87.0%

   \[\leadsto \color{blue}{\left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right) \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) + u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

10. Final simplification 87.0%

    \[\leadsto u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right) \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
11. Add Preprocessing

Alternative 7: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ t\_0 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right) + u2 \cdot \left(6.28318530718 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (+
    (*
     t_0
     (* (* u2 u2) (* u2 (fma u2 (* u2 81.6052492761019) -41.341702240407926))))
    (* u2 (* 6.28318530718 t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return (t_0 * ((u2 * u2) * (u2 * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f)))) + (u2 * (6.28318530718f * t_0));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(Float32(t_0 * Float32(Float32(u2 * u2) * Float32(u2 * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))))) + Float32(u2 * Float32(Float32(6.28318530718) * t_0)))
end

Derivation

1. Initial program 98.7%

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

   1. lift--.f32 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. neg-sub0 N/A

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

   10. lift--.f32 N/A

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

   11. associate--r- N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. lower-+.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-/.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   12. lower-*.f32 78.0

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

7. Applied rewrites 78.0%

   \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]

9. Applied rewrites 87.0%

   \[\leadsto \color{blue}{\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}} + u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

10. Final simplification 87.0%

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right) + u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
11. Add Preprocessing

Alternative 8: 89.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\frac{1}{u1 + -1}}{\frac{-1}{u1}}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ (/ 1.0 (+ u1 -1.0)) (/ -1.0 u1)))
  (*
   u2
   (+
    6.28318530718
    (* (* u2 u2) (fma u2 (* u2 81.6052492761019) -41.341702240407926))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((1.0f / (u1 + -1.0f)) / (-1.0f / u1))) * (u2 * (6.28318530718f + ((u2 * u2) * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(1.0) / Float32(u1 + Float32(-1.0))) / Float32(Float32(-1.0) / u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))))))
end

Derivation

1. Initial program 98.7%

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

   1. lift--.f32 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. neg-sub0 N/A

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

   10. lift--.f32 N/A

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

   11. associate--r- N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. lower-+.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-/.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   3. unpow2 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   12. lower-*.f32 78.0

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

7. Applied rewrites 78.0%

   \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(u2 \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right) + \frac{314159265359}{50000000000}\right)\right) \]

   2. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(u2 \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}\right) + \frac{314159265359}{50000000000}\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(u2 \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left(u2 \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(u2 \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

   8. lower-*.f32 86.9

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right)} + 6.28318530718\right)\right) \]

   9. lift-fma.f32 N/A

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

   12. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)} + \frac{314159265359}{50000000000}\right)\right) \]

   13. lower-*.f32 86.9

       \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot 81.6052492761019}, -41.341702240407926\right) + 6.28318530718\right)\right) \]

9. Applied rewrites 86.9%

   \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right) + 6.28318530718\right)}\right) \]

10. Final simplification 86.9%

    \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\frac{-1}{u1}}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right) \]
11. Add Preprocessing

Alternative 9: 86.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \left(u2 \cdot 81.6052492761019\right)\right) + \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 - -6.28318530718\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006500000134110451)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    (sqrt (fma u1 u1 u1))
    (*
     u2
     (+
      (* (* u2 u2) (* u2 (* u2 81.6052492761019)))
      (- (* (* u2 u2) -41.341702240407926) -6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006500000134110451f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * (u2 * (((u2 * u2) * (u2 * (u2 * 81.6052492761019f))) + (((u2 * u2) * -41.341702240407926f) - -6.28318530718f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.006500000134110451))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * Float32(u2 * Float32(Float32(Float32(u2 * u2) * Float32(u2 * Float32(u2 * Float32(81.6052492761019)))) + Float32(Float32(Float32(u2 * u2) * Float32(-41.341702240407926)) - Float32(-6.28318530718)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00650000013

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Applied rewrites 96.9%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 97.0

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

   7. Applied rewrites 97.0%

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

   if 0.00650000013 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.7%

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

      1. lift--.f32 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f32 N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-+.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-/.f32 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. lower-*.f32 44.2

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 44.2%

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot u1 + u1 \cdot 1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot u1 + \color{blue}{u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      4. lower-fma.f32 41.3

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   10. Applied rewrites 41.3%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   12. Applied rewrites 59.3%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \left(u2 \cdot 81.6052492761019\right)\right) + \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 - -6.28318530718\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \left(u2 \cdot 81.6052492761019\right)\right) + \left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 - -6.28318530718\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(6.28318530718 \cdot u2 + \left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006500000134110451)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    (sqrt (fma u1 u1 u1))
    (+
     (* 6.28318530718 u2)
     (*
      (* u2 u2)
      (* u2 (fma u2 (* u2 81.6052492761019) -41.341702240407926)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006500000134110451f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * ((6.28318530718f * u2) + ((u2 * u2) * (u2 * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.006500000134110451))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * Float32(Float32(Float32(6.28318530718) * u2) + Float32(Float32(u2 * u2) * Float32(u2 * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00650000013

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Applied rewrites 96.9%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 97.0

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

   7. Applied rewrites 97.0%

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

   if 0.00650000013 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.7%

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

      1. lift--.f32 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f32 N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-+.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-/.f32 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. lower-*.f32 44.2

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 44.2%

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot u1 + u1 \cdot 1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot u1 + \color{blue}{u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      4. lower-fma.f32 41.3

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   10. Applied rewrites 41.3%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   12. Applied rewrites 54.8%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right) + 6.28318530718 \cdot u2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(6.28318530718 \cdot u2 + \left(u2 \cdot u2\right) \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \left(u2 \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right) - -6.28318530718\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006500000134110451)
   (* 6.28318530718 (* u2 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    (sqrt (fma u1 u1 u1))
    (*
     u2
     (-
      (* u2 (* u2 (fma u2 (* u2 81.6052492761019) -41.341702240407926)))
      -6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006500000134110451f) {
		tmp = 6.28318530718f * (u2 * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * (u2 * ((u2 * (u2 * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f))) - -6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.006500000134110451))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * Float32(u2 * Float32(Float32(u2 * Float32(u2 * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926)))) - Float32(-6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00650000013

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Applied rewrites 96.9%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 97.0

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

   7. Applied rewrites 97.0%

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

   if 0.00650000013 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.7%

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

      1. lift--.f32 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f32 N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-+.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-/.f32 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)} + \frac{314159265359}{50000000000}\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}\right), \frac{314159265359}{50000000000}\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      12. lower-*.f32 44.2

          \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   7. Applied rewrites 44.2%

      \[\leadsto \sqrt{\frac{\frac{1}{u1 + -1}}{-\frac{1}{u1}}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot u1 + u1 \cdot 1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot u1 + \color{blue}{u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

      4. lower-fma.f32 41.3

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   10. Applied rewrites 41.3%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right) \]

   12. Applied rewrites 54.7%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \color{blue}{\left(u2 \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right) - -6.28318530718\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006500000134110451:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \left(u2 \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right)\right) - -6.28318530718\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-rgt-identity N/A

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

   18. lower-/.f32 N/A

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Applied rewrites 78.2%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 78.2

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

7. Applied rewrites 78.2%

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

8. Final simplification 78.2%

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

Alternative 13: 81.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-rgt-identity N/A

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

   18. lower-/.f32 N/A

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Applied rewrites 78.2%

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

Alternative 14: 64.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-rgt-identity N/A

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

   18. lower-/.f32 N/A

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Applied rewrites 78.2%

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

6. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 62.1

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

8. Applied rewrites 62.1%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))