Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 11.7s

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip3-- N/A

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

   4. associate-/r/ N/A

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

   5. associate-*l/ N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. distribute-rgt-in N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f32 N/A

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

   14. *-lft-identity N/A

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

   15. lower-fma.f32 N/A

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

   16. metadata-eval N/A

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

   17. sub-neg N/A

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

   18. cube-neg N/A

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ t_1 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.008999999612569809:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, \left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot u2, \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right)\right) \cdot t\_0, u2 \cdot u2, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))) (t_1 (cos (* u2 6.28318530718))))
   (if (<= (* t_0 t_1) 0.008999999612569809)
     (* (sqrt (fma (fma u1 u1 u1) u1 u1)) t_1)
     (fma
      (*
       (fma
        -85.45681720672748
        (* (* (* u2 u2) u2) u2)
        (fma 64.93939402268539 (* u2 u2) -19.739208802181317))
       t_0)
      (* u2 u2)
      t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float t_1 = cosf((u2 * 6.28318530718f));
	float tmp;
	if ((t_0 * t_1) <= 0.008999999612569809f) {
		tmp = sqrtf(fmaf(fmaf(u1, u1, u1), u1, u1)) * t_1;
	} else {
		tmp = fmaf((fmaf(-85.45681720672748f, (((u2 * u2) * u2) * u2), fmaf(64.93939402268539f, (u2 * u2), -19.739208802181317f)) * t_0), (u2 * u2), t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_1 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_1) <= Float32(0.008999999612569809))
		tmp = Float32(sqrt(fma(fma(u1, u1, u1), u1, u1)) * t_1);
	else
		tmp = fma(Float32(fma(Float32(-85.45681720672748), Float32(Float32(Float32(u2 * u2) * u2) * u2), fma(Float32(64.93939402268539), Float32(u2 * u2), Float32(-19.739208802181317))) * t_0), Float32(u2 * u2), t_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))) < 0.00899999961

   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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 98.9

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

   5. Applied rewrites 98.9%

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

   if 0.00899999961 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 97.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(u2 \cdot 6.28318530718\right) \leq 0.008999999612569809:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, \left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot u2, \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}}, u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ t_1 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.009499999694526196:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, \left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot u2, \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right)\right) \cdot t\_0, u2 \cdot u2, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))) (t_1 (cos (* u2 6.28318530718))))
   (if (<= (* t_0 t_1) 0.009499999694526196)
     (* (sqrt (fma u1 u1 u1)) t_1)
     (fma
      (*
       (fma
        -85.45681720672748
        (* (* (* u2 u2) u2) u2)
        (fma 64.93939402268539 (* u2 u2) -19.739208802181317))
       t_0)
      (* u2 u2)
      t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float t_1 = cosf((u2 * 6.28318530718f));
	float tmp;
	if ((t_0 * t_1) <= 0.009499999694526196f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * t_1;
	} else {
		tmp = fmaf((fmaf(-85.45681720672748f, (((u2 * u2) * u2) * u2), fmaf(64.93939402268539f, (u2 * u2), -19.739208802181317f)) * t_0), (u2 * u2), t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_1 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_1) <= Float32(0.009499999694526196))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * t_1);
	else
		tmp = fma(Float32(fma(Float32(-85.45681720672748), Float32(Float32(Float32(u2 * u2) * u2) * u2), fma(Float32(64.93939402268539), Float32(u2 * u2), Float32(-19.739208802181317))) * t_0), Float32(u2 * u2), t_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))) < 0.00949999969

   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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.00949999969 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 97.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(u2 \cdot 6.28318530718\right) \leq 0.009499999694526196:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, \left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot u2, \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}}, u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 5: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.2999999523162842:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, \left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot u2, \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right)\right) \cdot t\_0, u2 \cdot u2, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* u2 6.28318530718) 1.2999999523162842)
     (fma
      (*
       (fma
        -85.45681720672748
        (* (* (* u2 u2) u2) u2)
        (fma 64.93939402268539 (* u2 u2) -19.739208802181317))
       t_0)
      (* u2 u2)
      t_0)
     (* (sqrt u1) (cos (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((u2 * 6.28318530718f) <= 1.2999999523162842f) {
		tmp = fmaf((fmaf(-85.45681720672748f, (((u2 * u2) * u2) * u2), fmaf(64.93939402268539f, (u2 * u2), -19.739208802181317f)) * t_0), (u2 * u2), t_0);
	} else {
		tmp = sqrtf(u1) * cosf((u2 * 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(1.2999999523162842))
		tmp = fma(Float32(fma(Float32(-85.45681720672748), Float32(Float32(Float32(u2 * u2) * u2) * u2), fma(Float32(64.93939402268539), Float32(u2 * u2), Float32(-19.739208802181317))) * t_0), Float32(u2 * u2), t_0);
	else
		tmp = Float32(sqrt(u1) * cos(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1.29999995

   1. Initial program 99.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 98.8%

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

   if 1.29999995 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.7%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 79.1

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

   5. Applied rewrites 79.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.2999999523162842:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-85.45681720672748, \left(\left(u2 \cdot u2\right) \cdot u2\right) \cdot u2, \mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right)\right) \cdot \sqrt{\frac{u1}{1 - u1}}, u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 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 93.8%

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

6. Final simplification 93.8%

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

Alternative 7: 93.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 93.7

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

7. Applied rewrites 93.7%

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

8. Final simplification 93.7%

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

Alternative 8: 91.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

   1. lift-sqrt.f32 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. sqrt-prod N/A

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

   6. lower-*.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. lower-sqrt.f32 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 75.6

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

7. Applied rewrites 75.6%

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

8. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

10. Applied rewrites 91.8%

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

11. Final simplification 91.8%

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

Alternative 9: 91.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 91.7

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

7. Applied rewrites 91.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. times-frac N/A

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

   6. *-inverses N/A

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

   7. associate-/l* N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f32 91.7

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

9. Applied rewrites 91.7%

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

10. Final simplification 91.7%

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

Alternative 10: 84.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0017999999690800905:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right), u2 \cdot u2, 1\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0017999999690800905)
   (sqrt (/ u1 (- 1.0 u1)))
   (*
    (fma (fma 64.93939402268539 (* u2 u2) -19.739208802181317) (* u2 u2) 1.0)
    (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0017999999690800905f) {
		tmp = sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = fmaf(fmaf(64.93939402268539f, (u2 * u2), -19.739208802181317f), (u2 * u2), 1.0f) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0017999999690800905))
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	else
		tmp = Float32(fma(fma(Float32(64.93939402268539), Float32(u2 * u2), Float32(-19.739208802181317)), Float32(u2 * u2), Float32(1.0)) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00179999997

   1. Initial program 99.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 98.9%

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

   if 0.00179999997 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.3%

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

      1. lift-sqrt.f32 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. lower-sqrt.f32 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.6

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

   7. Applied rewrites 76.6%

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

   8. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 61.4

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

   10. Applied rewrites 61.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0017999999690800905:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(64.93939402268539, u2 \cdot u2, -19.739208802181317\right), u2 \cdot u2, 1\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. *-rgt-identity N/A

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

   10. sub-neg N/A

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

   11. rgt-mult-inverse N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. *-rgt-identity N/A

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

   16. distribute-lft-in N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. associate-*r* N/A

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

5. Applied rewrites 88.4%

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

Alternative 12: 79.6% 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 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 81.4%

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

Alternative 13: 73.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 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 81.4%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 75.3%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \]
9. Add Preprocessing

Alternative 14: 71.0% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 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 81.4%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1\right)} \]
7. Step-by-step derivation

8. Applied rewrites 72.7%

   \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \]
9. Add Preprocessing

Alternative 15: 62.8% accurate, 12.3× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

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 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 81.4%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
7. Step-by-step derivation

8. Applied rewrites 65.1%

   \[\leadsto \sqrt{u1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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))))