Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 13.7s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

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

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(fma(u1, u1, u1) / Float32(-fma(u1, u1, Float32(-1.0))))) * 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. 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. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. associate-*l/ N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

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

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

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

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

   19. sqr-neg N/A

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

   20. lower-+.f32 N/A

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

   21. 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-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 99.1

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

6. Applied rewrites 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.05000000074505806:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, t\_1 \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), t\_1\right)\\ \end{array} \end{array} \]

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) 0.05000000074505806)
     (* 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) <= 0.05000000074505806f) {
		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(0.05000000074505806))
		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))) < 0.0500000007

   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 0.0500000007 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 98.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 99.2%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.05000000074505806:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.016499999910593033:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, t\_1 \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), t\_1\right)\\ \end{array} \end{array} \]

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) 0.016499999910593033)
     (* t_0 (sqrt (fma 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) <= 0.016499999910593033f) {
		tmp = t_0 * sqrtf(fmaf(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(0.016499999910593033))
		tmp = Float32(t_0 * sqrt(fma(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))) < 0.0164999999

   1. Initial program 98.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.0164999999 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   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 99.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.016499999910593033:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2 \cdot u2, \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot -85.45681720672748, \mathsf{fma}\left(u2, u2 \cdot 64.93939402268539, -19.739208802181317\right)\right), \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return fma(fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0)), t_0, Float32(t_0 * Float32(fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539)) * Float32(u2 * Float32(u2 * Float32(u2 * 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. 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 81.8%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 94.5%

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

10. Applied rewrites 94.6%

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

Alternative 6: 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 94.5%

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

Alternative 7: 93.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * fmaf(fmaf(-85.45681720672748f, (u2 * u2), 64.93939402268539f), (u2 * (u2 * (u2 * u2))), 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(fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539)), Float32(u2 * Float32(u2 * Float32(u2 * u2))), 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

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
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 81.8%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 94.5%

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

10. Applied rewrites 94.5%

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

Alternative 8: 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 92.9%

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

Alternative 9: 88.3% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (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 * -19.739208802181317f), 1.0f);
}

Julia

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

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)} \]
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. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. *-rgt-identity N/A

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

   11. sub-neg N/A

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

   12. rgt-mult-inverse N/A

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

   13. mul-1-neg N/A

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

   14. distribute-neg-frac2 N/A

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

   15. mul-1-neg N/A

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

   16. *-rgt-identity N/A

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

   17. distribute-lft-in N/A

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

   18. +-commutative N/A

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

   19. sub-neg N/A

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

   20. associate-*r* N/A

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

5. Applied rewrites 90.7%

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

6. Final simplification 90.7%

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

Alternative 10: 88.3% 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 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 81.8%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} + \frac{-98696044010906577398881}{5000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)} \]
7. 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. lower-*.f32 N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. lft-mult-inverse N/A

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

   11. distribute-rgt-in N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. associate-/r* N/A

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

   15. *-rgt-identity N/A

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

   16. associate-*r/ N/A

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

   17. rgt-mult-inverse N/A

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

   18. lower-sqrt.f32 N/A

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

8. Applied rewrites 90.7%

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

9. Final simplification 90.7%

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

Alternative 11: 79.9% 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 81.8%

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

Alternative 12: 74.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(fma(u1, fma(u1, u1, u1), 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 81.8%

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

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 77.0%

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

Alternative 13: 71.6% 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 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 81.8%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 73.8%

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

Alternative 14: 63.2% 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 99.0%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 81.8%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 66.0%

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

Alternative 15: 19.2% accurate, 135.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = 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

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 81.8%

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

7. Applied rewrites 64.5%

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

8. Taylor expanded in u1 around inf

   \[\leadsto 1 \]
9. Step-by-step derivation

10. Applied rewrites 19.1%

    \[\leadsto 1 \]
11. Add Preprocessing

Alternative 16: 4.1% accurate, 135.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = 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

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 81.8%

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

7. Applied rewrites 64.5%

   \[\leadsto \sqrt{\frac{1}{\frac{u1 + 1}{u1}}} \]

8. Taylor expanded in u1 around -inf

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

10. Applied rewrites 3.9%

    \[\leadsto -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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))))