Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 14.0s

Alternatives: 16

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

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

   5. associate-*l/ N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. distribute-rgt-neg-in N/A

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

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

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

   19. sqr-neg N/A

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

   20. lower-+.f32 N/A

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

   21. lower-*.f32 98.2

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

4. Applied rewrites 98.2%

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

5. Final simplification 98.2%

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

Alternative 2: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.007499999832361937:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, \frac{1}{\sqrt{\frac{1 - u1}{u1}}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{t\_0} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.007499999832361937)
     (* (sin (* 6.28318530718 u2)) (sqrt (fma u1 (fma u1 u1 u1) u1)))
     (*
      u2
      (fma
       6.28318530718
       (/ 1.0 (sqrt (/ (- 1.0 u1) u1)))
       (*
        (* u2 u2)
        (*
         (sqrt t_0)
         (fma
          (* u2 u2)
          (fma u2 (* u2 -76.70585975309672) 81.6052492761019)
          -41.341702240407926))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.007499999832361937f) {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = u2 * fmaf(6.28318530718f, (1.0f / sqrtf(((1.0f - u1) / u1))), ((u2 * u2) * (sqrtf(t_0) * fmaf((u2 * u2), fmaf(u2, (u2 * -76.70585975309672f), 81.6052492761019f), -41.341702240407926f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.007499999832361937))
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = Float32(u2 * fma(Float32(6.28318530718), Float32(Float32(1.0) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1))), Float32(Float32(u2 * u2) * Float32(sqrt(t_0) * fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019)), Float32(-41.341702240407926))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00749999983

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.00749999983 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.2%

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

   3. Taylor expanded in 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. Applied rewrites 95.3%

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

   7. Applied rewrites 95.8%

      \[\leadsto u2 \cdot \mathsf{fma}\left(6.28318530718, \frac{1}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.007499999832361937:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(6.28318530718, \frac{1}{\sqrt{\frac{1 - u1}{u1}}}, \left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 98.2%

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

3. Final simplification 98.2%

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

Alternative 4: 93.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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. Applied rewrites 94.3%

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

7. Applied rewrites 94.3%

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

Alternative 5: 93.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (/ (fma u1 u1 u1) (+ -1.0 (* u1 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(-(fmaf(u1, u1, u1) / (-1.0f + (u1 * 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(-Float32(fma(u1, u1, u1) / Float32(Float32(-1.0) + Float32(u1 * u1))))) * Float32(u2 * fma(u2, Float32(u2 * fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)), Float32(-41.341702240407926))), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.2%

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

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

   5. associate-*l/ N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. distribute-rgt-neg-in N/A

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

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

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

   19. sqr-neg N/A

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

   20. lower-+.f32 N/A

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

   21. lower-*.f32 98.2

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f32 N/A

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

7. Applied rewrites 94.3%

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

8. Final simplification 94.3%

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

Alternative 6: 93.9% accurate, 2.0× speedup

Math

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

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(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019)), Float32(-41.341702240407926)), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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. metadata-eval N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f32 N/A

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

   16. lower-*.f32 94.2

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

5. Applied rewrites 94.2%

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

Alternative 7: 87.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.009449999779462814:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1 - u1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.009449999779462814)
   (*
    (sqrt (fma u1 (fma u1 u1 u1) u1))
    (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718)))
   (/ (* 6.28318530718 u2) (sqrt (/ (- 1.0 u1) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.009449999779462814f) {
		tmp = sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1)) * (u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	} else {
		tmp = (6.28318530718f * u2) / sqrtf(((1.0f - u1) / u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.009449999779462814))
		tmp = Float32(sqrt(fma(u1, fma(u1, u1, u1), u1)) * Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(Float32(6.28318530718) * u2) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00944999978

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 88.1

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

   8. Applied rewrites 88.1%

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

   if 0.00944999978 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.1%

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

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

      5. associate-*l/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f32 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. distribute-rgt-neg-in N/A

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

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

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

      19. sqr-neg N/A

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

      20. lower-+.f32 N/A

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

      21. lower-*.f32 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 87.1

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

   7. Applied rewrites 87.1%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. sqrt-div N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f32 N/A

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

      8. remove-double-neg N/A

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

      9. lift-fma.f32 N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-/l/ N/A

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

      14. lift-+.f32 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f32 N/A

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

      20. flip-- N/A

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

      21. lift--.f32 N/A

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

   9. Applied rewrites 86.6%

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

       1. lift-*.f32 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f32 N/A

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

       4. lift-sqrt.f32 N/A

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

       5. lift-/.f32 N/A

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

       6. sqrt-div N/A

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

       7. metadata-eval N/A

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

       8. lift-sqrt.f32 N/A

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

       9. associate-/r/ N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f32 N/A

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

       12. lift-sqrt.f32 N/A

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

       13. lift-sqrt.f32 N/A

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

       14. sqrt-undiv N/A

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

       15. lower-sqrt.f32 N/A

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

       16. lower-/.f32 87.0

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

   11. Applied rewrites 87.0%

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

Alternative 8: 86.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.0007999999797903001)
     (*
      (sqrt (fma u1 u1 u1))
      (* u2 (fma u2 (* u2 -41.341702240407926) 6.28318530718)))
     (* (sqrt t_0) (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.0007999999797903001f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * (u2 * fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f));
	} else {
		tmp = sqrtf(t_0) * (6.28318530718f * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0007999999797903001))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * Float32(u2 * fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718))));
	else
		tmp = Float32(sqrt(t_0) * Float32(Float32(6.28318530718) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 7.9999998e-4

   1. Initial program 98.2%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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. flip3-- N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. associate-/r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{{1}^{3} - {u1}^{3}} \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      5. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}{{1}^{3} - {u1}^{3}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}{{1}^{3} - {u1}^{3}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{u1 \cdot \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)}{{1}^{3} - {u1}^{3}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\frac{u1 \cdot \color{blue}{\left(\left(u1 \cdot u1 + 1 \cdot u1\right) + 1\right)}}{{1}^{3} - {u1}^{3}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \sqrt{\frac{\color{blue}{u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + u1 \cdot 1}}{{1}^{3} - {u1}^{3}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \sqrt{\frac{u1 \cdot \left(u1 \cdot u1 + 1 \cdot u1\right) + \color{blue}{u1}}{{1}^{3} - {u1}^{3}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      11. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot u1 + 1 \cdot u1, u1\right)}}{{1}^{3} - {u1}^{3}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      12. *-lft-identity N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1, u1 \cdot u1 + \color{blue}{u1}, u1\right)}{{1}^{3} - {u1}^{3}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}, u1\right)}{{1}^{3} - {u1}^{3}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}{\color{blue}{1} - {u1}^{3}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      15. sub-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}{\color{blue}{1 + \left(\mathsf{neg}\left({u1}^{3}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      16. cube-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}{1 + \color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}{\color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3} + 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f32 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 88.4

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

   10. Applied rewrites 88.4%

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

   if 7.9999998e-4 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.2%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-*.f32 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 9: 91.8% accurate, 2.4× speedup

Math

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

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(u2, Float32(u2 * fma(u2, Float32(u2 * Float32(81.6052492761019)), Float32(-41.341702240407926))), Float32(6.28318530718))))
end

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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. Applied rewrites 92.0%

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

Alternative 10: 89.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 82.2

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

5. Applied rewrites 82.2%

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

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. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f32 N/A

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

8. Applied rewrites 89.1%

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

Alternative 11: 76.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005210000090301037:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \left(\mathsf{fma}\left(u1, 0.5, 1\right) \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.005210000090301037)
   (* (* 6.28318530718 u2) (* (fma u1 0.5 1.0) (sqrt 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) <= 0.005210000090301037f) {
		tmp = (6.28318530718f * u2) * (fmaf(u1, 0.5f, 1.0f) * sqrtf(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(0.005210000090301037))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * Float32(fma(u1, Float32(0.5), Float32(1.0)) * sqrt(u1)));
	else
		tmp = Float32(Float32(u2 * 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) < 0.00521000009

   1. Initial program 98.5%

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

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

      5. associate-*l/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f32 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. distribute-rgt-neg-in N/A

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

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

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

      19. sqr-neg N/A

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

      20. lower-+.f32 N/A

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

      21. lower-*.f32 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 97.7

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

   7. Applied rewrites 97.7%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. sqrt-div N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f32 N/A

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

      8. remove-double-neg N/A

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

      9. lift-fma.f32 N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-/l/ N/A

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

      14. lift-+.f32 N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f32 N/A

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

      20. flip-- N/A

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

      21. lift--.f32 N/A

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

   9. Applied rewrites 97.4%

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

   10. Taylor expanded in u1 around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f32 84.3

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(u1, 0.5, 1\right)} \cdot \sqrt{u1}\right) \cdot \left(6.28318530718 \cdot u2\right) \]

   12. Applied rewrites 84.3%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(u1, 0.5, 1\right)} \cdot \sqrt{u1}\right) \cdot \left(6.28318530718 \cdot u2\right) \]

   if 0.00521000009 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-*.f32 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 45.8

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in u2 around 0

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

       1. lower-*.f32 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-fma.f32 N/A

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

       7. lower-*.f32 56.9

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

   11. Applied rewrites 56.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005210000090301037:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \left(\mathsf{fma}\left(u1, 0.5, 1\right) \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005210000090301037:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.005210000090301037)
   (* (* 6.28318530718 u2) (sqrt (fma 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) <= 0.005210000090301037f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(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(0.005210000090301037))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, u1, u1)));
	else
		tmp = Float32(Float32(u2 * 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) < 0.00521000009

   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

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

      1. lower-*.f32 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f32 84.0

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

   8. Applied rewrites 84.0%

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

   if 0.00521000009 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u2 around 0

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

      1. lower-*.f32 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 45.8

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in u2 around 0

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

       1. lower-*.f32 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-fma.f32 N/A

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

       7. lower-*.f32 56.9

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

   11. Applied rewrites 56.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005210000090301037:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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. Applied rewrites 94.3%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 82.2%

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

Alternative 14: 75.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 82.2

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. lower-fma.f32 75.7

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

8. Applied rewrites 75.7%

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

Alternative 15: 73.0% 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 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 82.2

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f32 72.4

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

8. Applied rewrites 72.4%

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

9. Final simplification 72.4%

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

Alternative 16: 64.7% accurate, 6.4× speedup

Math

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

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(Float32(6.28318530718) * u2) * sqrt(u1))
end

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 82.2

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

5. Applied rewrites 82.2%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 64.5

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

8. Applied rewrites 64.5%

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

9. Final simplification 64.5%

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

Reproduce

herbie shell --seed 2024223 
(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))))