Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 11.0s

Alternatives: 17

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin \left(6.28318530718 \cdot u2\right) \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (sin (* 6.28318530718 u2)) 0.009999999776482582)
   (* u2 (* 6.28318530718 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    (fma
     u2
     (* u2 (fma u2 (* u2 81.6052492761019) -41.341702240407926))
     6.28318530718)
    (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (sinf((6.28318530718f * u2)) <= 0.009999999776482582f) {
		tmp = u2 * (6.28318530718f * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = fmaf(u2, (u2 * fmaf(u2, (u2 * 81.6052492761019f), -41.341702240407926f)), 6.28318530718f) * (u2 * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (sin(Float32(Float32(6.28318530718) * u2)) <= Float32(0.009999999776482582))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(fma(u2, Float32(u2 * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))), Float32(6.28318530718)) * Float32(u2 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (sin.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.00999999978

   1. Initial program 98.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. lower-sqrt.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 94.9

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

   5. Simplified 94.9%

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

      1. lift-*.f32 N/A

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

      2. *-lft-identity N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. inv-pow N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lift-/.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f32 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. +-commutative N/A

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

      18. lift-+.f32 N/A

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

      19. sqrt-div N/A

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

   7. Applied egg-rr 95.0%

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

   if 0.00999999978 < (sin.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 98.1%

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

   3. Taylor expanded in 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 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-*.f32 71.1

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

   8. Simplified 71.1%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-fma.f32 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 71.1

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

      10. lift-fma.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 71.1

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 71.1

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

   10. Applied egg-rr 71.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(6.28318530718 \cdot u2\right) \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin \left(6.28318530718 \cdot u2\right) \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (sin (* 6.28318530718 u2)) 0.009999999776482582)
   (* u2 (* 6.28318530718 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    (sqrt u1)
    (*
     u2
     (fma
      (* u2 u2)
      (fma u2 (* u2 81.6052492761019) -41.341702240407926)
      6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (sinf((6.28318530718f * u2)) <= 0.009999999776482582f) {
		tmp = u2 * (6.28318530718f * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = sqrtf(u1) * (u2 * fmaf((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 (sin(Float32(Float32(6.28318530718) * u2)) <= Float32(0.009999999776482582))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * fma(Float32(u2 * 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 (sin.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.00999999978

   1. Initial program 98.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. lower-sqrt.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 94.9

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

   5. Simplified 94.9%

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

      1. lift-*.f32 N/A

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

      2. *-lft-identity N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. inv-pow N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lift-/.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f32 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. +-commutative N/A

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

      18. lift-+.f32 N/A

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

      19. sqrt-div N/A

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

   7. Applied egg-rr 95.0%

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

   if 0.00999999978 < (sin.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 98.1%

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

   3. Taylor expanded in 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 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-*.f32 71.1

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

   8. Simplified 71.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(6.28318530718 \cdot u2\right) \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot 81.6052492761019, -41.341702240407926\right), 6.28318530718\right)\right)\\ \end{array} \]
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 98.4%

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

3. Final simplification 98.4%

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

Alternative 5: 93.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ u2 \cdot \mathsf{fma}\left(t\_0 \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    u2
    (fma
     (* t_0 (* u2 u2))
     (fma
      u2
      (* u2 (fma (* u2 u2) -76.70585975309672 81.6052492761019))
      -41.341702240407926)
     (* 6.28318530718 t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return u2 * fmaf((t_0 * (u2 * u2)), fmaf(u2, (u2 * fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f)), -41.341702240407926f), (6.28318530718f * t_0));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(u2 * fma(Float32(t_0 * Float32(u2 * u2)), fma(u2, Float32(u2 * fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019))), Float32(-41.341702240407926)), Float32(Float32(6.28318530718) * t_0)))
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)\right)} \]

5. Simplified 95.3%

   \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
6. Add Preprocessing

Alternative 6: 93.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (*
   u2
   (fma
    (* u2 u2)
    (fma
     u2
     (* u2 (fma (* u2 u2) -76.70585975309672 81.6052492761019))
     -41.341702240407926)
    6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * fmaf((u2 * u2), fmaf(u2, (u2 * fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f)), -41.341702240407926f), 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * fma(Float32(u2 * u2), fma(u2, Float32(u2 * fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019))), Float32(-41.341702240407926)), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right) + \frac{314159265359}{50000000000}\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right), \frac{314159265359}{50000000000}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, u2 \cdot \left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right) + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right)\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}, \frac{314159265359}{50000000000}\right)\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   14. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right)}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \frac{314159265359}{50000000000}\right)\right) \]

   16. lower-*.f32 95.2

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right) \]

5. Simplified 95.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)\right)} \]
6. Add Preprocessing

Alternative 7: 91.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 93.6%

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

Alternative 8: 84.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (* u2 (* 6.28318530718 (sqrt (/ u1 (- 1.0 u1)))))
   (* u2 (* (fma u2 (* u2 -41.341702240407926) 6.28318530718) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = u2 * (6.28318530718f * sqrtf((u1 / (1.0f - u1))));
	} else {
		tmp = u2 * (fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.009999999776482582))
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(u2 * Float32(fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)) * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. lower-sqrt.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 96.8

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

   5. Simplified 96.8%

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

      1. lift-*.f32 N/A

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

      2. *-lft-identity N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. inv-pow N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. lift-/.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f32 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

      17. +-commutative N/A

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

      18. lift-+.f32 N/A

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

      19. sqrt-div N/A

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

   7. Applied egg-rr 96.9%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 79.1

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

   5. Simplified 79.1%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 62.1

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

   8. Simplified 62.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (* u2 (* (fma u2 (* u2 -41.341702240407926) 6.28318530718) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = u2 * (fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.009999999776482582))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(u2 * Float32(fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)) * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. lower-sqrt.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 96.8

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

   5. Simplified 96.8%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 79.1

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

   5. Simplified 79.1%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 62.1

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

   8. Simplified 62.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-out N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. lower-*.f32 N/A

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

5. Simplified 91.6%

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

Alternative 11: 78.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (* (* 6.28318530718 u2) (sqrt (fma u1 (fma u1 u1 u1) u1)))
   (* u2 (* (fma u2 (* u2 -41.341702240407926) 6.28318530718) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = u2 * (fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.009999999776482582))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = Float32(u2 * Float32(fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)) * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. lower-sqrt.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-lft1-in N/A

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

      8. lower-fma.f32 89.9

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

   8. Simplified 89.9%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 79.1

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

   5. Simplified 79.1%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 62.1

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

   8. Simplified 62.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \left(\sqrt{u1} \cdot \left(6.28318530718 + u1 \cdot 3.14159265359\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (* u2 (* (sqrt u1) (+ 6.28318530718 (* u1 3.14159265359))))
   (* u2 (* (fma u2 (* u2 -41.341702240407926) 6.28318530718) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = u2 * (sqrtf(u1) * (6.28318530718f + (u1 * 3.14159265359f)));
	} else {
		tmp = u2 * (fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.009999999776482582))
		tmp = Float32(u2 * Float32(sqrt(u1) * Float32(Float32(6.28318530718) + Float32(u1 * Float32(3.14159265359)))));
	else
		tmp = Float32(u2 * Float32(fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)) * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. lower-sqrt.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 96.8

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

   5. Simplified 96.8%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{100000000000} \cdot \left(\sqrt{{u1}^{3}} \cdot u2\right) + \frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}}\right) \cdot u2} + \frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}}\right) \cdot u2 + \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \cdot u2} \]

      3. distribute-rgt-out N/A

         \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

      5. *-commutative N/A

         \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{{u1}^{3}} \cdot \frac{314159265359}{100000000000}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{{u1}^{3}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

      7. lower-sqrt.f32 N/A

         \[\leadsto u2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{{u1}^{3}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

      8. cube-mult N/A

         \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{\color{blue}{u1 \cdot \left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

      9. unpow2 N/A

         \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{{u1}^{2}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

      10. lower-*.f32 N/A

          \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{\color{blue}{u1 \cdot {u1}^{2}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

      11. unpow2 N/A

          \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

      13. *-commutative N/A

          \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \frac{314159265359}{100000000000}, \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \frac{314159265359}{100000000000}, \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

      15. lower-sqrt.f32 86.6

          \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 3.14159265359, \color{blue}{\sqrt{u1}} \cdot 6.28318530718\right) \]

   8. Simplified 86.6%

      \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 3.14159265359, \sqrt{u1} \cdot 6.28318530718\right)} \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

      3. lift-sqrt.f32 N/A

         \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

      4. lift-sqrt.f32 N/A

         \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000} + \color{blue}{\sqrt{u1}} \cdot \frac{314159265359}{50000000000}\right) \]

      5. lift-*.f32 N/A

         \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000} + \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

      6. +-commutative N/A

         \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000}\right)} \]

      7. lift-*.f32 N/A

         \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}} + \sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000}\right) \]

      8. lift-sqrt.f32 N/A

         \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \color{blue}{\sqrt{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000}\right) \]

      9. lift-*.f32 N/A

         \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000}\right) \]

      10. sqrt-prod N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1 \cdot u1}\right)} \cdot \frac{314159265359}{100000000000}\right) \]

      11. lift-sqrt.f32 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \left(\color{blue}{\sqrt{u1}} \cdot \sqrt{u1 \cdot u1}\right) \cdot \frac{314159265359}{100000000000}\right) \]

      12. pow1/2 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \left(\sqrt{u1} \cdot \color{blue}{{\left(u1 \cdot u1\right)}^{\frac{1}{2}}}\right) \cdot \frac{314159265359}{100000000000}\right) \]

      13. associate-*l* N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \color{blue}{\sqrt{u1} \cdot \left({\left(u1 \cdot u1\right)}^{\frac{1}{2}} \cdot \frac{314159265359}{100000000000}\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + {\left(u1 \cdot u1\right)}^{\frac{1}{2}} \cdot \frac{314159265359}{100000000000}\right)\right)} \]

      15. lower-*.f32 N/A

          \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + {\left(u1 \cdot u1\right)}^{\frac{1}{2}} \cdot \frac{314159265359}{100000000000}\right)\right)} \]

      16. lower-+.f32 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} + {\left(u1 \cdot u1\right)}^{\frac{1}{2}} \cdot \frac{314159265359}{100000000000}\right)}\right) \]

      17. pow1/2 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \color{blue}{\sqrt{u1 \cdot u1}} \cdot \frac{314159265359}{100000000000}\right)\right) \]

      18. lift-*.f32 N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \sqrt{\color{blue}{u1 \cdot u1}} \cdot \frac{314159265359}{100000000000}\right)\right) \]

      19. sqrt-prod N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1}\right)} \cdot \frac{314159265359}{100000000000}\right)\right) \]

      20. rem-square-sqrt N/A

          \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \color{blue}{u1} \cdot \frac{314159265359}{100000000000}\right)\right) \]

   10. Applied egg-rr 86.6%

       \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(6.28318530718 + u1 \cdot 3.14159265359\right)\right)} \]

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 79.1

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

   5. Simplified 79.1%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 62.1

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

   8. Simplified 62.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;u2 \cdot \left(\sqrt{u1} \cdot \left(6.28318530718 + u1 \cdot 3.14159265359\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{u1} \cdot \left(6.28318530718 + u1 \cdot 3.14159265359\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (* (sqrt u1) (+ 6.28318530718 (* u1 3.14159265359)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf(u1) * (6.28318530718f + (u1 * 3.14159265359f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u2 * (sqrt(u1) * (6.28318530718e0 + (u1 * 3.14159265359e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(u1) * Float32(Float32(6.28318530718) + Float32(u1 * Float32(3.14159265359)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (sqrt(u1) * (single(6.28318530718) + (u1 * single(3.14159265359))));
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 84.3

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

5. Simplified 84.3%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{100000000000} \cdot \left(\sqrt{{u1}^{3}} \cdot u2\right) + \frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}}\right) \cdot u2} + \frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}}\right) \cdot u2 + \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \cdot u2} \]

   3. distribute-rgt-out N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

   5. *-commutative N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{{u1}^{3}} \cdot \frac{314159265359}{100000000000}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   6. lower-fma.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{{u1}^{3}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

   7. lower-sqrt.f32 N/A

      \[\leadsto u2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{{u1}^{3}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   8. cube-mult N/A

      \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{\color{blue}{u1 \cdot \left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   9. unpow2 N/A

      \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{{u1}^{2}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   10. lower-*.f32 N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{\color{blue}{u1 \cdot {u1}^{2}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   11. unpow2 N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   12. lower-*.f32 N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   13. *-commutative N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \frac{314159265359}{100000000000}, \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

   14. lower-*.f32 N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \frac{314159265359}{100000000000}, \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

   15. lower-sqrt.f32 76.6

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 3.14159265359, \color{blue}{\sqrt{u1}} \cdot 6.28318530718\right) \]

8. Simplified 76.6%

   \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 3.14159265359, \sqrt{u1} \cdot 6.28318530718\right)} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

   2. lift-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

   4. lift-sqrt.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000} + \color{blue}{\sqrt{u1}} \cdot \frac{314159265359}{50000000000}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000} + \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

   6. +-commutative N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000}\right)} \]

   7. lift-*.f32 N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}} + \sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000}\right) \]

   8. lift-sqrt.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \color{blue}{\sqrt{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000}\right) \]

   9. lift-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000}\right) \]

   10. sqrt-prod N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1 \cdot u1}\right)} \cdot \frac{314159265359}{100000000000}\right) \]

   11. lift-sqrt.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \left(\color{blue}{\sqrt{u1}} \cdot \sqrt{u1 \cdot u1}\right) \cdot \frac{314159265359}{100000000000}\right) \]

   12. pow1/2 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \left(\sqrt{u1} \cdot \color{blue}{{\left(u1 \cdot u1\right)}^{\frac{1}{2}}}\right) \cdot \frac{314159265359}{100000000000}\right) \]

   13. associate-*l* N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000} + \color{blue}{\sqrt{u1} \cdot \left({\left(u1 \cdot u1\right)}^{\frac{1}{2}} \cdot \frac{314159265359}{100000000000}\right)}\right) \]

   14. distribute-lft-out N/A

       \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + {\left(u1 \cdot u1\right)}^{\frac{1}{2}} \cdot \frac{314159265359}{100000000000}\right)\right)} \]

   15. lower-*.f32 N/A

       \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + {\left(u1 \cdot u1\right)}^{\frac{1}{2}} \cdot \frac{314159265359}{100000000000}\right)\right)} \]

   16. lower-+.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} + {\left(u1 \cdot u1\right)}^{\frac{1}{2}} \cdot \frac{314159265359}{100000000000}\right)}\right) \]

   17. pow1/2 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \color{blue}{\sqrt{u1 \cdot u1}} \cdot \frac{314159265359}{100000000000}\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \sqrt{\color{blue}{u1 \cdot u1}} \cdot \frac{314159265359}{100000000000}\right)\right) \]

   19. sqrt-prod N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \color{blue}{\left(\sqrt{u1} \cdot \sqrt{u1}\right)} \cdot \frac{314159265359}{100000000000}\right)\right) \]

   20. rem-square-sqrt N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\frac{314159265359}{50000000000} + \color{blue}{u1} \cdot \frac{314159265359}{100000000000}\right)\right) \]

10. Applied egg-rr 76.6%

    \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(6.28318530718 + u1 \cdot 3.14159265359\right)\right)} \]
11. Add Preprocessing

Alternative 14: 72.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(u1, 3.14159265359, 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (* (sqrt u1) (fma u1 3.14159265359 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf(u1) * fmaf(u1, 3.14159265359f, 6.28318530718f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(u1) * fma(u1, Float32(3.14159265359), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 84.3

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

5. Simplified 84.3%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{100000000000} \cdot \left(\sqrt{{u1}^{3}} \cdot u2\right) + \frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}}\right) \cdot u2} + \frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}}\right) \cdot u2 + \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \cdot u2} \]

   3. distribute-rgt-out N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{100000000000} \cdot \sqrt{{u1}^{3}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

   5. *-commutative N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{{u1}^{3}} \cdot \frac{314159265359}{100000000000}} + \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   6. lower-fma.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{{u1}^{3}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right)} \]

   7. lower-sqrt.f32 N/A

      \[\leadsto u2 \cdot \mathsf{fma}\left(\color{blue}{\sqrt{{u1}^{3}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   8. cube-mult N/A

      \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{\color{blue}{u1 \cdot \left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   9. unpow2 N/A

      \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{{u1}^{2}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   10. lower-*.f32 N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{\color{blue}{u1 \cdot {u1}^{2}}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   11. unpow2 N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   12. lower-*.f32 N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}}, \frac{314159265359}{100000000000}, \frac{314159265359}{50000000000} \cdot \sqrt{u1}\right) \]

   13. *-commutative N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \frac{314159265359}{100000000000}, \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

   14. lower-*.f32 N/A

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \frac{314159265359}{100000000000}, \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

   15. lower-sqrt.f32 76.6

       \[\leadsto u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 3.14159265359, \color{blue}{\sqrt{u1}} \cdot 6.28318530718\right) \]

8. Simplified 76.6%

   \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 3.14159265359, \sqrt{u1} \cdot 6.28318530718\right)} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \color{blue}{\left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

   2. lift-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1 \cdot \left(u1 \cdot u1\right)}} \cdot \frac{314159265359}{100000000000} + \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \]

   4. lift-sqrt.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000} + \color{blue}{\sqrt{u1}} \cdot \frac{314159265359}{50000000000}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)} \cdot \frac{314159265359}{100000000000} + \color{blue}{\sqrt{u1} \cdot \frac{314159265359}{50000000000}}\right) \]

   6. lift-fma.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \frac{314159265359}{100000000000}, \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right)} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \frac{314159265359}{100000000000}, \sqrt{u1} \cdot \frac{314159265359}{50000000000}\right) \cdot u2} \]

   8. lower-*.f32 76.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 3.14159265359, \sqrt{u1} \cdot 6.28318530718\right) \cdot u2} \]

10. Applied egg-rr 76.6%

    \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot \mathsf{fma}\left(u1, 3.14159265359, 6.28318530718\right)\right) \cdot u2} \]

11. Final simplification 76.6%

    \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(u1, 3.14159265359, 6.28318530718\right)\right) \]
12. Add Preprocessing

Alternative 15: 72.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 84.3

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

5. Simplified 84.3%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f32 76.2

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

8. Simplified 76.2%

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

Alternative 16: 64.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (6.28318530718f * 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 = u2 * (6.28318530718e0 * sqrt(u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 84.3

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

5. Simplified 84.3%

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

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. lower-*.f32 N/A

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

   3. lower-sqrt.f32 67.7

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

8. Simplified 67.7%

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

   1. lift-sqrt.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 67.8

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

10. Applied egg-rr 67.8%

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

11. Final simplification 67.8%

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

Alternative 17: 64.3% 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 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 84.3

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

5. Simplified 84.3%

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

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. lower-*.f32 N/A

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

   3. lower-sqrt.f32 67.7

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

8. Simplified 67.7%

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

9. Final simplification 67.7%

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

Reproduce

herbie shell --seed 2024219 
(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))))