Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 12.5s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-(fmaf(u1, u1, u1) / (-1.0f + (u1 * u1)))) * cosf((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))))) * cos(Float32(Float32(6.28318530718) * u2)))
end

Derivation

1. Initial program 99.0%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. *-lft-identity N/A

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

   10. lower-fma.f32 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

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

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

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

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

   17. sqr-neg N/A

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

   18. lower-+.f32 N/A

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

   19. lower-*.f32 99.1

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.07339999824762344:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, t\_1 \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))) (t_1 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 t_1) 0.07339999824762344)
     (* t_0 (sqrt (fma u1 (fma u1 u1 u1) u1)))
     (fma
      (* u2 u2)
      (*
       t_1
       (fma
        (* u2 u2)
        (* (* u2 u2) -85.45681720672748)
        (fma u2 (* u2 64.93939402268539) -19.739208802181317)))
      t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float t_1 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((t_0 * t_1) <= 0.07339999824762344f) {
		tmp = t_0 * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = fmaf((u2 * u2), (t_1 * fmaf((u2 * u2), ((u2 * u2) * -85.45681720672748f), fmaf(u2, (u2 * 64.93939402268539f), -19.739208802181317f))), t_1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	t_1 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_1) <= Float32(0.07339999824762344))
		tmp = Float32(t_0 * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = fma(Float32(u2 * u2), Float32(t_1 * fma(Float32(u2 * u2), Float32(Float32(u2 * u2) * Float32(-85.45681720672748)), fma(u2, Float32(u2 * Float32(64.93939402268539)), Float32(-19.739208802181317)))), t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 0.0733999982

   1. Initial program 99.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.0733999982 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 98.8%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), \sqrt{\frac{u1}{1 - u1}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.07339999824762344:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.10999999940395355:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (cos (* 6.28318530718 u2)) (sqrt (/ u1 (- 1.0 u1))))
      0.10999999940395355)
   (*
    (sqrt (fma u1 (fma u1 u1 u1) u1))
    (fma u2 (* u2 -19.739208802181317) 1.0))
   (* (sqrt (* u1 (/ 1.0 (- 1.0 u1)))) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((cosf((6.28318530718f * u2)) * sqrtf((u1 / (1.0f - u1)))) <= 0.10999999940395355f) {
		tmp = sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1)) * fmaf(u2, (u2 * -19.739208802181317f), 1.0f);
	} else {
		tmp = sqrtf((u1 * (1.0f / (1.0f - u1)))) * 1.0f;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))) <= Float32(0.10999999940395355))
		tmp = Float32(sqrt(fma(u1, fma(u1, u1, u1), u1)) * fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 0.109999999

   1. Initial program 99.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \color{blue}{\left(1 + \frac{-98696044010906577398881}{5000000000000000000000} \cdot {u2}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \frac{-98696044010906577398881}{5000000000000000000000}, 1\right)} \]

      6. lower-*.f32 89.4

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot -19.739208802181317}, 1\right) \]

   8. Applied rewrites 89.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)} \]

   if 0.109999999 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 98.8%

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

   4. Applied rewrites 98.9%

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

      1. lift-+.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-fma.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. sub-div N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \color{blue}{1} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.10999999940395355:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 5: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathsf{fma}\left(u2 \cdot u2, t\_0 \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (fma
    (* u2 u2)
    (*
     t_0
     (fma
      (* u2 u2)
      (* (* u2 u2) -85.45681720672748)
      (fma u2 (* u2 64.93939402268539) -19.739208802181317)))
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return fmaf((u2 * u2), (t_0 * fmaf((u2 * u2), ((u2 * u2) * -85.45681720672748f), fmaf(u2, (u2 * 64.93939402268539f), -19.739208802181317f))), t_0);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return fma(Float32(u2 * u2), Float32(t_0 * fma(Float32(u2 * u2), Float32(Float32(u2 * u2) * Float32(-85.45681720672748)), fma(u2, Float32(u2 * Float32(64.93939402268539)), Float32(-19.739208802181317)))), t_0)
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right)} \]

5. Applied rewrites 95.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), \sqrt{\frac{u1}{1 - u1}}\right)} \]
6. Add Preprocessing

Alternative 6: 93.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\frac{\mathsf{fma}\left(u1, u1, u1\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (/ (fma u1 u1 u1) (+ -1.0 (* u1 u1)))))
  (fma
   (* u2 u2)
   (fma
    (* u2 u2)
    (fma u2 (* u2 -85.45681720672748) 64.93939402268539)
    -19.739208802181317)
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-(fmaf(u1, u1, u1) / (-1.0f + (u1 * u1)))) * fmaf((u2 * u2), fmaf((u2 * u2), fmaf(u2, (u2 * -85.45681720672748f), 64.93939402268539f), -19.739208802181317f), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(fma(u1, u1, u1) / Float32(Float32(-1.0) + Float32(u1 * u1))))) * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-85.45681720672748)), Float32(64.93939402268539)), Float32(-19.739208802181317)), Float32(1.0)))
end

Derivation

1. Initial program 99.0%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. *-lft-identity N/A

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

   10. lower-fma.f32 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

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

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

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

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

   17. sqr-neg N/A

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

   18. lower-+.f32 N/A

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

   19. lower-*.f32 99.1

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right)} \]

   2. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}, 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}, 1\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}, 1\right) \]

   5. sub-neg N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)}, 1\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}, 1\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-98696044010906577398881}{5000000000000000000000}\right)}, 1\right) \]

   8. unpow2 N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}, \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right) \]

   10. +-commutative N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}}, \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right) \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{u2}^{2} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right) \]

   12. unpow2 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right) \]

   13. associate-*l* N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{u2 \cdot \left(u2 \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}\right)} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right) \]

   14. lower-fma.f32 N/A

       \[\leadsto \sqrt{\frac{\mathsf{neg}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}\right)}, \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right) \]

   15. lower-*.f32 95.5

       \[\leadsto \sqrt{\frac{-\mathsf{fma}\left(u1, u1, u1\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, \color{blue}{u2 \cdot -85.45681720672748}, 64.93939402268539\right), -19.739208802181317\right), 1\right) \]

7. Applied rewrites 95.5%

   \[\leadsto \sqrt{\frac{-\mathsf{fma}\left(u1, u1, u1\right)}{-1 + u1 \cdot u1}} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)} \]

8. Final simplification 95.5%

   \[\leadsto \sqrt{-\frac{\mathsf{fma}\left(u1, u1, u1\right)}{-1 + u1 \cdot u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \]
9. Add Preprocessing

Alternative 7: 93.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{-1 + \frac{1}{u1}}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ 1.0 (+ -1.0 (/ 1.0 u1))))
  (fma
   (* u2 u2)
   (fma
    u2
    (* u2 (fma (* u2 u2) -85.45681720672748 64.93939402268539))
    -19.739208802181317)
   1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((1.0f / (-1.0f + (1.0f / u1)))) * fmaf((u2 * u2), fmaf(u2, (u2 * fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f)), -19.739208802181317f), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)))) * fma(Float32(u2 * u2), fma(u2, Float32(u2 * fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539))), Float32(-19.739208802181317)), Float32(1.0)))
end

Derivation

1. Initial program 99.0%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. *-lft-identity N/A

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

   10. lower-fma.f32 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

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

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

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

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

   17. sqr-neg N/A

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

   18. lower-+.f32 N/A

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

   19. lower-*.f32 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. clear-num N/A

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

   6. frac-2neg N/A

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

   7. lift-neg.f32 N/A

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

   8. remove-double-neg N/A

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

   9. lift-fma.f32 N/A

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

   10. distribute-lft1-in N/A

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

   11. lift-+.f32 N/A

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

   12. *-commutative N/A

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

   13. associate-/l/ N/A

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

   14. lift-+.f32 N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

   19. lift-*.f32 N/A

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

   20. lift-+.f32 N/A

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

   21. +-commutative N/A

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

   22. flip-- N/A

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

   23. lift--.f32 N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]

9. Applied rewrites 95.3%

   \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)} \]

10. Final simplification 95.3%

    \[\leadsto \sqrt{\frac{1}{-1 + \frac{1}{u1}}} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \]
11. Add Preprocessing

Alternative 8: 93.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \cdot \sqrt{u1 \cdot \frac{1}{1 - u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (fma
   (* u2 u2)
   (fma
    u2
    (* u2 (fma (* u2 u2) -85.45681720672748 64.93939402268539))
    -19.739208802181317)
   1.0)
  (sqrt (* u1 (/ 1.0 (- 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf((u2 * u2), fmaf(u2, (u2 * fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f)), -19.739208802181317f), 1.0f) * sqrtf((u1 * (1.0f / (1.0f - u1))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(Float32(u2 * u2), fma(u2, Float32(u2 * fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539))), Float32(-19.739208802181317)), Float32(1.0)) * sqrt(Float32(u1 * Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))))
end

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

   1. lift-+.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. sub-div N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]

9. Applied rewrites 95.3%

   \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)} \]

10. Final simplification 95.3%

    \[\leadsto \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right) \cdot \sqrt{u1 \cdot \frac{1}{1 - u1}} \]
11. Add Preprocessing

Alternative 9: 91.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot 64.93939402268539, \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (fma
   (* u2 u2)
   (* (* u2 u2) 64.93939402268539)
   (fma u2 (* u2 -19.739208802181317) 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf((u2 * u2), ((u2 * u2) * 64.93939402268539f), fmaf(u2, (u2 * -19.739208802181317f), 1.0f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(u2 * u2), Float32(Float32(u2 * u2) * Float32(64.93939402268539)), fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0))))
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-98696044010906577398881}{5000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)\right)} \]

5. Applied rewrites 93.6%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot 64.93939402268539, \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\right)} \]
6. Add Preprocessing

Alternative 10: 91.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right), 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* u1 (/ 1.0 (- 1.0 u1))))
  (fma u2 (* u2 (fma (* u2 u2) 64.93939402268539 -19.739208802181317)) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f / (1.0f - u1)))) * fmaf(u2, (u2 * fmaf((u2 * u2), 64.93939402268539f, -19.739208802181317f)), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 * Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * fma(u2, Float32(u2 * fma(Float32(u2 * u2), Float32(64.93939402268539), Float32(-19.739208802181317))), Float32(1.0)))
end

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

   1. lift-+.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. sub-div N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \color{blue}{\left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} + 1\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right), 1\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)}, 1\right) \]

   6. sub-neg N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} + \left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right)}, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\color{blue}{{u2}^{2} \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}} + \left(\mathsf{neg}\left(\frac{98696044010906577398881}{5000000000000000000000}\right)\right)\right), 1\right) \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \mathsf{fma}\left(u2, u2 \cdot \left({u2}^{2} \cdot \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \color{blue}{\frac{-98696044010906577398881}{5000000000000000000000}}\right), 1\right) \]

   9. lower-fma.f32 N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \frac{-98696044010906577398881}{5000000000000000000000}\right)}, 1\right) \]

   10. unpow2 N/A

       \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000}, \frac{-98696044010906577398881}{5000000000000000000000}\right), 1\right) \]

   11. lower-*.f32 93.5

       \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, 64.93939402268539, -19.739208802181317\right), 1\right) \]

9. Applied rewrites 93.5%

   \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right), 1\right)} \]

10. Final simplification 93.5%

    \[\leadsto \sqrt{u1 \cdot \frac{1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right), 1\right) \]
11. Add Preprocessing

Alternative 11: 88.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (fma -19.739208802181317 (* u2 u2) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf(-19.739208802181317f, (u2 * u2), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)))
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f32 91.4

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

5. Applied rewrites 91.4%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 77.0%

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

9. Taylor expanded in u2 around 0

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

    1. *-rgt-identity N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. distribute-lft-out N/A

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

    6. lower-*.f32 N/A

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

    7. lower-sqrt.f32 N/A

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

    8. lower-/.f32 N/A

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

    9. lower--.f32 N/A

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

    10. +-commutative N/A

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

    11. lower-fma.f32 N/A

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

    12. unpow2 N/A

        \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\frac{-98696044010906577398881}{5000000000000000000000}, \color{blue}{u2 \cdot u2}, 1\right) \]

    13. lower-*.f32 90.5

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

11. Applied rewrites 90.5%

    \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)} \]
12. Add Preprocessing

Alternative 12: 80.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 83.0%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
6. Add Preprocessing

Alternative 13: 63.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sqrt{u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f * 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 = 1.0e0 * sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 73.8%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\frac{1}{\sqrt{u1 + 1}}}{\frac{1}{\sqrt{u1}}} \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 64.2%

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

8. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot 1 \]
9. Step-by-step derivation

   1. lower-sqrt.f32 65.4

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot 1 \]

10. Applied rewrites 65.4%

    \[\leadsto \color{blue}{\sqrt{u1}} \cdot 1 \]

11. Final simplification 65.4%

    \[\leadsto 1 \cdot \sqrt{u1} \]
12. Add Preprocessing

Alternative 14: 19.3% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 1.0 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f * 1.0f;
}

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 = 1.0e0 * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) * Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) * single(1.0);
end

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 73.8%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\frac{1}{\sqrt{u1 + 1}}}{\frac{1}{\sqrt{u1}}} \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 64.2%

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

8. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{1} \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 19.6%

    \[\leadsto \color{blue}{1} \cdot 1 \]

11. Final simplification 19.6%

    \[\leadsto 1 \cdot 1 \]
12. Add Preprocessing

Alternative 15: 4.0% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot -1 \end{array} \]

FPCore

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

C

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

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 = 1.0e0 * (-1.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 73.8%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \frac{\frac{1}{\sqrt{u1 + 1}}}{\frac{1}{\sqrt{u1}}} \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 64.2%

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

8. Taylor expanded in u1 around -inf

   \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot 1 \]
9. Step-by-step derivation

   1. unpow2 N/A

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

   2. rem-square-sqrt 3.5

      \[\leadsto \color{blue}{-1} \cdot 1 \]

10. Applied rewrites 3.5%

    \[\leadsto \color{blue}{-1} \cdot 1 \]

11. Final simplification 3.5%

    \[\leadsto 1 \cdot -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))