Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 11.5s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 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

Derivation

1. Initial program 99.1%

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

Alternative 2: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.04500000178813934:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.04500000178813934)
     (* (sqrt (fma u1 u1 u1)) (fma u2 (* u2 -19.739208802181317) 1.0))
     t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((t_0 * cosf((6.28318530718f * u2))) <= 0.04500000178813934f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * fmaf(u2, (u2 * -19.739208802181317f), 1.0f);
	} else {
		tmp = t_0;
	}
	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(t_0 * cos(Float32(Float32(6.28318530718) * u2))) <= Float32(0.04500000178813934))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0)));
	else
		tmp = 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))) < 0.0450000018

   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 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-*.f32 88.3

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

   8. Applied rewrites 88.3%

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

   if 0.0450000018 < (*.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

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   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 89.1%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.6000000238418579:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), \left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.6000000238418579)
     (fma
      t_0
      (fma (* u2 u2) -19.739208802181317 1.0)
      (*
       (* u2 u2)
       (*
        (* u2 u2)
        (* t_0 (fma -85.45681720672748 (* u2 u2) 64.93939402268539)))))
     (* (cos (* 6.28318530718 u2)) (sqrt (fma u1 (fma u1 u1 u1) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 0.6000000238418579f) {
		tmp = fmaf(t_0, fmaf((u2 * u2), -19.739208802181317f, 1.0f), ((u2 * u2) * ((u2 * u2) * (t_0 * fmaf(-85.45681720672748f, (u2 * u2), 64.93939402268539f)))));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	}
	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(Float32(6.28318530718) * u2) <= Float32(0.6000000238418579))
		tmp = fma(t_0, fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)), Float32(Float32(u2 * u2) * Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539))))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.600000024

   1. Initial program 99.4%

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

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. neg-sub0 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{0 - \left(1 - u1\right)}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{1}{0 - \color{blue}{\left(1 - u1\right)}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      9. associate--r- N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\left(0 - 1\right) + u1}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-neg.f32 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f32 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lift--.f32 N/A

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

      15. div-inv N/A

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

      16. lift-neg.f32 N/A

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

      17. frac-2neg N/A

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

      18. sqrt-div N/A

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

      19. lift-sqrt.f32 N/A

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

      20. lower-/.f32 N/A

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

   6. Applied rewrites 99.0%

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

   7. 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. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

   9. Applied rewrites 99.4%

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

   if 0.600000024 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.6000000238418579:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}}, \mathsf{fma}\left(u2 \cdot u2, -19.739208802181317, 1\right), \left(u2 \cdot u2\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(-85.45681720672748, u2 \cdot u2, 64.93939402268539\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 1.0f) {
		tmp = fmaf(t_0, fmaf((u2 * u2), -19.739208802181317f, 1.0f), ((u2 * u2) * ((u2 * u2) * (t_0 * fmaf(-85.45681720672748f, (u2 * u2), 64.93939402268539f)))));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(fmaf(u1, u1, u1));
	}
	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(Float32(6.28318530718) * u2) <= Float32(1.0))
		tmp = fma(t_0, fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)), Float32(Float32(u2 * u2) * Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539))))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1

   1. Initial program 99.4%

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

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. neg-sub0 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{0 - \left(1 - u1\right)}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      8. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{1}{0 - \color{blue}{\left(1 - u1\right)}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      9. associate--r- N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\left(0 - 1\right) + u1}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-neg.f32 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f32 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lift--.f32 N/A

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

      15. div-inv N/A

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

      16. lift-neg.f32 N/A

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

      17. frac-2neg N/A

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

      18. sqrt-div N/A

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

      19. lift-sqrt.f32 N/A

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

      20. lower-/.f32 N/A

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

   6. Applied rewrites 99.0%

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

   7. 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. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

   9. Applied rewrites 99.3%

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

   if 1 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 95.8%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 82.6

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

   5. Applied rewrites 82.6%

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

4. Final simplification 98.0%

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

Alternative 5: 93.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	return fma(t_0, fma(Float32(u2 * u2), Float32(-19.739208802181317), Float32(1.0)), Float32(Float32(u2 * u2) * Float32(Float32(u2 * u2) * Float32(t_0 * fma(Float32(-85.45681720672748), Float32(u2 * u2), Float32(64.93939402268539))))))
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. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. neg-sub0 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{0 - \left(1 - u1\right)}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. lift--.f32 N/A

      \[\leadsto \sqrt{\frac{1}{0 - \color{blue}{\left(1 - u1\right)}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. associate--r- N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\left(0 - 1\right) + u1}} \cdot \left(\mathsf{neg}\left(u1\right)\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. lift--.f32 N/A

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

   15. div-inv N/A

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

   16. lift-neg.f32 N/A

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

   17. frac-2neg N/A

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

   18. sqrt-div N/A

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

   19. lift-sqrt.f32 N/A

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

   20. lower-/.f32 N/A

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

6. Applied rewrites 98.8%

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

7. 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. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

9. Applied rewrites 94.0%

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

Alternative 6: 93.5% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (fma
    (* t_0 (* u2 u2))
    (fma
     (* u2 u2)
     (fma u2 (* u2 -85.45681720672748) 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((t_0 * (u2 * u2)), fmaf((u2 * u2), fmaf(u2, (u2 * -85.45681720672748f), 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(t_0 * Float32(u2 * u2)), fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(-85.45681720672748)), Float32(64.93939402268539)), Float32(-19.739208802181317)), 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 94.0%

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

Alternative 7: 93.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 93.9

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

5. Applied rewrites 93.9%

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

Alternative 8: 91.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(u2 * u2), fma(u2, Float32(u2 * Float32(64.93939402268539)), Float32(-19.739208802181317)), 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. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-*.f32 92.9

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

5. Applied rewrites 92.9%

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

Alternative 9: 88.2% 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 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 90.4

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

5. Applied rewrites 90.4%

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

Alternative 10: 88.2% 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 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 83.3%

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

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. +-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. *-rgt-identity N/A

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

   13. sub-neg N/A

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

   14. rgt-mult-inverse N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-frac2 N/A

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

   17. mul-1-neg N/A

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

   18. *-rgt-identity N/A

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

   19. distribute-lft-in N/A

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

   20. +-commutative N/A

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

   21. sub-neg N/A

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

8. Applied rewrites 90.4%

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

9. Final simplification 90.4%

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

Alternative 11: 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 83.3%

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

Alternative 12: 74.0% 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 99.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
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 83.3%

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

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 76.3%

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

Alternative 13: 71.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u1, 0.5, 1\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (fma u1 0.5 1.0) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf(u1, 0.5f, 1.0f) * sqrtf(u1);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(u1, Float32(0.5), Float32(1.0)) * 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 83.3%

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

7. Applied rewrites 82.7%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 73.5%

    \[\leadsto \mathsf{fma}\left(u1, 0.5, 1\right) \cdot \sqrt{\color{blue}{u1}} \]
11. Add Preprocessing

Alternative 14: 71.2% 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 83.3%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 73.4%

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

Alternative 15: 63.1% 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 83.3%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 64.3%

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

Reproduce

herbie shell --seed 2024233 
(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))))