Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.7s

Alternatives: 9

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, 0.9× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ u2\_s \cdot \left(\sqrt{\frac{u1 \cdot \mathsf{fma}\left(u1, u1, 1\right)}{\mathsf{fma}\left(u1, u1, 1\right) \cdot \left(1 - u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\_m\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (*
  u2_s
  (*
   (sqrt (/ (* u1 (fma u1 u1 1.0)) (* (fma u1 u1 1.0) (- 1.0 u1))))
   (sin (* 6.28318530718 u2_m)))))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	return u2_s * (sqrtf(((u1 * fmaf(u1, u1, 1.0f)) / (fmaf(u1, u1, 1.0f) * (1.0f - u1)))) * sinf((6.28318530718f * u2_m)));
}

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	return Float32(u2_s * Float32(sqrt(Float32(Float32(u1 * fma(u1, u1, Float32(1.0))) / Float32(fma(u1, u1, Float32(1.0)) * Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(6.28318530718) * u2_m))))
end

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.0%

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

Alternative 2: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ u2\_s \cdot \begin{array}{l} \mathbf{if}\;u2\_m \leq 0.041999999433755875:\\ \;\;\;\;\sqrt{u1 \cdot \frac{-1}{-1 + u1}} \cdot \left(\left(\left(\left(\left(\mathsf{fma}\left(u2\_m \cdot u2\_m, -76.70585975309672, 81.6052492761019\right) \cdot u2\_m\right) \cdot u2\_m - 41.341702240407926\right) \cdot u2\_m\right) \cdot u2\_m + 6.28318530718\right) \cdot u2\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + u1\right) \cdot u1} \cdot \sin \left(6.28318530718 \cdot u2\_m\right)\\ \end{array} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (*
  u2_s
  (if (<= u2_m 0.041999999433755875)
    (*
     (sqrt (* u1 (/ -1.0 (+ -1.0 u1))))
     (*
      (+
       (*
        (*
         (-
          (*
           (* (fma (* u2_m u2_m) -76.70585975309672 81.6052492761019) u2_m)
           u2_m)
          41.341702240407926)
         u2_m)
        u2_m)
       6.28318530718)
      u2_m))
    (* (sqrt (* (+ 1.0 u1) u1)) (sin (* 6.28318530718 u2_m))))))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	float tmp;
	if (u2_m <= 0.041999999433755875f) {
		tmp = sqrtf((u1 * (-1.0f / (-1.0f + u1)))) * (((((((fmaf((u2_m * u2_m), -76.70585975309672f, 81.6052492761019f) * u2_m) * u2_m) - 41.341702240407926f) * u2_m) * u2_m) + 6.28318530718f) * u2_m);
	} else {
		tmp = sqrtf(((1.0f + u1) * u1)) * sinf((6.28318530718f * u2_m));
	}
	return u2_s * tmp;
}

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	tmp = Float32(0.0)
	if (u2_m <= Float32(0.041999999433755875))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(-1.0) / Float32(Float32(-1.0) + u1)))) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(fma(Float32(u2_m * u2_m), Float32(-76.70585975309672), Float32(81.6052492761019)) * u2_m) * u2_m) - Float32(41.341702240407926)) * u2_m) * u2_m) + Float32(6.28318530718)) * u2_m));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(1.0) + u1) * u1)) * sin(Float32(Float32(6.28318530718) * u2_m)));
	end
	return Float32(u2_s * tmp)
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0419999994

   1. Initial program 98.6%

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 90.0%

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

   9. Applied rewrites 98.4%

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

   if 0.0419999994 < u2

   1. Initial program 97.8%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. frac-2neg N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sub-sign-inv N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. sqr-neg-rev N/A

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

      13. lower-+.f32 N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f32 N/A

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

      16. distribute-neg-in N/A

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

      17. *-rgt-identity N/A

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

      18. distribute-lft-neg-in N/A

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

      19. fp-cancel-sub-sign N/A

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

      20. *-rgt-identity N/A

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

      21. lower--.f32 N/A

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

      22. metadata-eval 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f32 N/A

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

      5. *-lft-identity N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. distribute-neg-in N/A

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

      11. distribute-frac-neg2 N/A

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

      12. distribute-frac-neg N/A

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

      13. lift-+.f32 N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f32 N/A

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

      17. distribute-lft-neg-out N/A

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

      18. fp-cancel-sub-sign-inv N/A

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

      19. metadata-eval N/A

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

      20. flip-- N/A

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

      21. *-lft-identity N/A

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

      22. lift--.f32 N/A

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in u1 around 0

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

      1. lower-+.f32 87.8

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

   9. Applied rewrites 87.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.041999999433755875:\\ \;\;\;\;\sqrt{u1 \cdot \frac{-1}{-1 + u1}} \cdot \left(\left(\left(\left(\left(\mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right) \cdot u2\right) \cdot u2 - 41.341702240407926\right) \cdot u2\right) \cdot u2 + 6.28318530718\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + u1\right) \cdot u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ u2\_s \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\_m\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (* u2_s (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2_m)))))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	return u2_s * (sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2_m)));
}

Fortran

cosTheta_i_m = abs(costheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0d0, u2)
real(4) function code(u2_s, costheta_i_m, u1, u2_m)
    real(4), intent (in) :: u2_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2_m
    code = u2_s * (sqrt((u1 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2_m)))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	return Float32(u2_s * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2_m))))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
u2\_m = abs(u2);
u2\_s = sign(u2) * abs(1.0);
function tmp = code(u2_s, cosTheta_i_m, u1, u2_m)
	tmp = u2_s * (sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2_m)));
end

Derivation

1. Initial program 98.5%

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

Alternative 4: 94.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ u2\_s \cdot \begin{array}{l} \mathbf{if}\;u2\_m \leq 0.041999999433755875:\\ \;\;\;\;\sqrt{u1 \cdot \frac{-1}{-1 + u1}} \cdot \left(\left(\left(\left(\left(\mathsf{fma}\left(u2\_m \cdot u2\_m, -76.70585975309672, 81.6052492761019\right) \cdot u2\_m\right) \cdot u2\_m - 41.341702240407926\right) \cdot u2\_m\right) \cdot u2\_m + 6.28318530718\right) \cdot u2\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\_m\right)\\ \end{array} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (*
  u2_s
  (if (<= u2_m 0.041999999433755875)
    (*
     (sqrt (* u1 (/ -1.0 (+ -1.0 u1))))
     (*
      (+
       (*
        (*
         (-
          (*
           (* (fma (* u2_m u2_m) -76.70585975309672 81.6052492761019) u2_m)
           u2_m)
          41.341702240407926)
         u2_m)
        u2_m)
       6.28318530718)
      u2_m))
    (* (sqrt u1) (sin (* 6.28318530718 u2_m))))))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	float tmp;
	if (u2_m <= 0.041999999433755875f) {
		tmp = sqrtf((u1 * (-1.0f / (-1.0f + u1)))) * (((((((fmaf((u2_m * u2_m), -76.70585975309672f, 81.6052492761019f) * u2_m) * u2_m) - 41.341702240407926f) * u2_m) * u2_m) + 6.28318530718f) * u2_m);
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2_m));
	}
	return u2_s * tmp;
}

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	tmp = Float32(0.0)
	if (u2_m <= Float32(0.041999999433755875))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(-1.0) / Float32(Float32(-1.0) + u1)))) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(fma(Float32(u2_m * u2_m), Float32(-76.70585975309672), Float32(81.6052492761019)) * u2_m) * u2_m) - Float32(41.341702240407926)) * u2_m) * u2_m) + Float32(6.28318530718)) * u2_m));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2_m)));
	end
	return Float32(u2_s * tmp)
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0419999994

   1. Initial program 98.6%

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 90.0%

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

   9. Applied rewrites 98.4%

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

   if 0.0419999994 < u2

   1. Initial program 97.8%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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-sqrt.f32 73.5

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

   5. Applied rewrites 73.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.041999999433755875:\\ \;\;\;\;\sqrt{u1 \cdot \frac{-1}{-1 + u1}} \cdot \left(\left(\left(\left(\left(\mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right) \cdot u2\right) \cdot u2 - 41.341702240407926\right) \cdot u2\right) \cdot u2 + 6.28318530718\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ u2\_s \cdot \left(\sqrt{u1 \cdot \frac{-1}{-1 + u1}} \cdot \left(\left(\left(\left(\left(\mathsf{fma}\left(u2\_m \cdot u2\_m, -76.70585975309672, 81.6052492761019\right) \cdot u2\_m\right) \cdot u2\_m - 41.341702240407926\right) \cdot u2\_m\right) \cdot u2\_m + 6.28318530718\right) \cdot u2\_m\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (*
  u2_s
  (*
   (sqrt (* u1 (/ -1.0 (+ -1.0 u1))))
   (*
    (+
     (*
      (*
       (-
        (*
         (* (fma (* u2_m u2_m) -76.70585975309672 81.6052492761019) u2_m)
         u2_m)
        41.341702240407926)
       u2_m)
      u2_m)
     6.28318530718)
    u2_m))))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	return u2_s * (sqrtf((u1 * (-1.0f / (-1.0f + u1)))) * (((((((fmaf((u2_m * u2_m), -76.70585975309672f, 81.6052492761019f) * u2_m) * u2_m) - 41.341702240407926f) * u2_m) * u2_m) + 6.28318530718f) * u2_m));
}

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	return Float32(u2_s * Float32(sqrt(Float32(u1 * Float32(Float32(-1.0) / Float32(Float32(-1.0) + u1)))) * Float32(Float32(Float32(Float32(Float32(Float32(Float32(fma(Float32(u2_m * u2_m), Float32(-76.70585975309672), Float32(81.6052492761019)) * u2_m) * u2_m) - Float32(41.341702240407926)) * u2_m) * u2_m) + Float32(6.28318530718)) * u2_m)))
end

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 78.7%

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

9. Applied rewrites 89.1%

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

10. Final simplification 89.1%

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

Alternative 6: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ u2\_s \cdot \left(\left(t\_0 \cdot \left(-41.341702240407926 \cdot \left(u2\_m \cdot u2\_m\right)\right) - -6.28318530718 \cdot t\_0\right) \cdot u2\_m\right) \end{array} \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (*
    u2_s
    (*
     (- (* t_0 (* -41.341702240407926 (* u2_m u2_m))) (* -6.28318530718 t_0))
     u2_m))))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return u2_s * (((t_0 * (-41.341702240407926f * (u2_m * u2_m))) - (-6.28318530718f * t_0)) * u2_m);
}

Fortran

cosTheta_i_m = abs(costheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0d0, u2)
real(4) function code(u2_s, costheta_i_m, u1, u2_m)
    real(4), intent (in) :: u2_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2_m
    real(4) :: t_0
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    code = u2_s * (((t_0 * ((-41.341702240407926e0) * (u2_m * u2_m))) - ((-6.28318530718e0) * t_0)) * u2_m)
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return Float32(u2_s * Float32(Float32(Float32(t_0 * Float32(Float32(-41.341702240407926) * Float32(u2_m * u2_m))) - Float32(Float32(-6.28318530718) * t_0)) * u2_m))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
u2\_m = abs(u2);
u2\_s = sign(u2) * abs(1.0);
function tmp = code(u2_s, cosTheta_i_m, u1, u2_m)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	tmp = u2_s * (((t_0 * (single(-41.341702240407926) * (u2_m * u2_m))) - (single(-6.28318530718) * t_0)) * u2_m);
end

Derivation

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 78.8

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

5. Applied rewrites 78.8%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 78.6%

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

10. Applied rewrites 87.0%

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

Alternative 7: 81.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ u2\_s \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2\_m\right)\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (* u2_s (* (sqrt (/ u1 (- 1.0 u1))) (* 6.28318530718 u2_m))))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	return u2_s * (sqrtf((u1 / (1.0f - u1))) * (6.28318530718f * u2_m));
}

Fortran

cosTheta_i_m = abs(costheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0d0, u2)
real(4) function code(u2_s, costheta_i_m, u1, u2_m)
    real(4), intent (in) :: u2_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2_m
    code = u2_s * (sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 * u2_m))
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	return Float32(u2_s * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) * u2_m)))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
u2\_m = abs(u2);
u2\_s = sign(u2) * abs(1.0);
function tmp = code(u2_s, cosTheta_i_m, u1, u2_m)
	tmp = u2_s * (sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) * u2_m));
end

Derivation

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 78.8

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

5. Applied rewrites 78.8%

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

Alternative 8: 65.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ u2\_s \cdot \left(\left(\sqrt{u1} \cdot u2\_m\right) \cdot 6.28318530718\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (* u2_s (* (* (sqrt u1) u2_m) 6.28318530718)))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	return u2_s * ((sqrtf(u1) * u2_m) * 6.28318530718f);
}

Fortran

cosTheta_i_m = abs(costheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0d0, u2)
real(4) function code(u2_s, costheta_i_m, u1, u2_m)
    real(4), intent (in) :: u2_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2_m
    code = u2_s * ((sqrt(u1) * u2_m) * 6.28318530718e0)
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	return Float32(u2_s * Float32(Float32(sqrt(u1) * u2_m) * Float32(6.28318530718)))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
u2\_m = abs(u2);
u2\_s = sign(u2) * abs(1.0);
function tmp = code(u2_s, cosTheta_i_m, u1, u2_m)
	tmp = u2_s * ((sqrt(u1) * u2_m) * single(6.28318530718));
end

Derivation

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 78.8

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

5. Applied rewrites 78.8%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 61.4%

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

Alternative 9: 65.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} cosTheta_i_m = \left|cosTheta\_i\right| \\ u2\_m = \left|u2\right| \\ u2\_s = \mathsf{copysign}\left(1, u2\right) \\ u2\_s \cdot \left(\left(\sqrt{u1} \cdot 6.28318530718\right) \cdot u2\_m\right) \end{array} \]

FPCore

cosTheta_i_m = (fabs.f32 cosTheta_i)
u2\_m = (fabs.f32 u2)
u2\_s = (copysign.f32 #s(literal 1 binary32) u2)
(FPCore (u2_s cosTheta_i_m u1 u2_m)
 :precision binary32
 (* u2_s (* (* (sqrt u1) 6.28318530718) u2_m)))

C

cosTheta_i_m = fabs(cosTheta_i);
u2\_m = fabs(u2);
u2\_s = copysign(1.0, u2);
float code(float u2_s, float cosTheta_i_m, float u1, float u2_m) {
	return u2_s * ((sqrtf(u1) * 6.28318530718f) * u2_m);
}

Fortran

cosTheta_i_m = abs(costheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0d0, u2)
real(4) function code(u2_s, costheta_i_m, u1, u2_m)
    real(4), intent (in) :: u2_s
    real(4), intent (in) :: costheta_i_m
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2_m
    code = u2_s * ((sqrt(u1) * 6.28318530718e0) * u2_m)
end function

Julia

cosTheta_i_m = abs(cosTheta_i)
u2\_m = abs(u2)
u2\_s = copysign(1.0, u2)
function code(u2_s, cosTheta_i_m, u1, u2_m)
	return Float32(u2_s * Float32(Float32(sqrt(u1) * Float32(6.28318530718)) * u2_m))
end

MATLAB

cosTheta_i_m = abs(cosTheta_i);
u2\_m = abs(u2);
u2\_s = sign(u2) * abs(1.0);
function tmp = code(u2_s, cosTheta_i_m, u1, u2_m)
	tmp = u2_s * ((sqrt(u1) * single(6.28318530718)) * u2_m);
end

Derivation

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 78.8

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

5. Applied rewrites 78.8%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 61.4%

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

10. Applied rewrites 61.4%

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

Reproduce

herbie shell --seed 2024337 
(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))))