Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 12.9s

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 99.0%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ t_1 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.08500000089406967:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))) (t_1 (cos (* 6.28318530718 u2))))
   (if (<= (* t_0 t_1) 0.08500000089406967)
     (* t_1 (sqrt (fma u1 (fma u1 u1 u1) 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)));
	float t_1 = cosf((6.28318530718f * u2));
	float tmp;
	if ((t_0 * t_1) <= 0.08500000089406967f) {
		tmp = t_1 * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = fmaf((t_0 * (u2 * u2)), fmaf((u2 * u2), fmaf(u2, (u2 * -85.45681720672748f), 64.93939402268539f), -19.739208802181317f), t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_1 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_1) <= Float32(0.08500000089406967))
		tmp = Float32(t_1 * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = 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
	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))) < 0.0850000009

   1. Initial program 98.9%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.0850000009 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.1%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.08500000089406967:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ t_1 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.01600000075995922:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))) (t_1 (cos (* 6.28318530718 u2))))
   (if (<= (* t_0 t_1) 0.01600000075995922)
     (* t_1 (sqrt (fma u1 u1 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)));
	float t_1 = cosf((6.28318530718f * u2));
	float tmp;
	if ((t_0 * t_1) <= 0.01600000075995922f) {
		tmp = t_1 * sqrtf(fmaf(u1, u1, u1));
	} else {
		tmp = fmaf((t_0 * (u2 * u2)), fmaf((u2 * u2), fmaf(u2, (u2 * -85.45681720672748f), 64.93939402268539f), -19.739208802181317f), t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_1 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_1) <= Float32(0.01600000075995922))
		tmp = Float32(t_1 * sqrt(fma(u1, u1, u1)));
	else
		tmp = 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
	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))) < 0.0160000008

   1. Initial program 98.8%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.0160000008 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.2%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 96.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.01600000075995922:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.05999999865889549:\\ \;\;\;\;\mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.05999999865889549)
     (*
      (fma -19.739208802181317 (* u2 u2) 1.0)
      (sqrt (fma u1 (fma u1 u1 u1) u1)))
     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))) <= 0.05999999865889549f) {
		tmp = fmaf(-19.739208802181317f, (u2 * u2), 1.0f) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} 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(0.05999999865889549))
		tmp = Float32(fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	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))) < 0.0599999987

   1. Initial program 98.9%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. lift-/.f32 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. lower-pow.f32 N/A

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

      8. lift--.f32 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

          \[\leadsto {\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      11. *-inverses N/A

          \[\leadsto {\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. metadata-eval 98.9

          \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f32 N/A

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

      17. sub-neg N/A

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

      18. distribute-lft-in N/A

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

      19. metadata-eval N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

   7. Applied rewrites 87.0%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 87.2%

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

   if 0.0599999987 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.1%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   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 87.4%

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

Alternative 5: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.026000000536441803:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.026000000536441803)
     (* (sqrt (fma u1 u1 u1)) (fma -19.739208802181317 (* u2 u2) 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))) <= 0.026000000536441803f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * fmaf(-19.739208802181317f, (u2 * u2), 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(0.026000000536441803))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * fma(Float32(-19.739208802181317), Float32(u2 * u2), 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))) < 0.0260000005

   1. Initial program 98.8%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 98.0

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

   5. Applied rewrites 98.0%

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

   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. lower-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 86.5

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

   8. Applied rewrites 86.5%

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

   if 0.0260000005 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.2%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 87.6%

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

Alternative 6: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.999986469745636:\\ \;\;\;\;\mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.999986469745636)
   (* (fma -19.739208802181317 (* u2 u2) 1.0) (sqrt u1))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.999986469745636f) {
		tmp = fmaf(-19.739208802181317f, (u2 * u2), 1.0f) * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.999986469745636))
		tmp = Float32(fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)) * sqrt(u1));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99998647

   1. Initial program 97.6%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. lift-/.f32 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. lower-pow.f32 N/A

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

      8. lift--.f32 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

          \[\leadsto {\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      11. *-inverses N/A

          \[\leadsto {\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. metadata-eval 97.7

          \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. Applied rewrites 97.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f32 N/A

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

      17. sub-neg N/A

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

      18. distribute-lft-in N/A

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

      19. metadata-eval N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{fma}\left(\frac{-98696044010906577398881}{5000000000000000000000}, u2 \cdot u2, 1\right) \cdot \sqrt{u1} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.0%

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

   if 0.99998647 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.6%

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

   3. Taylor expanded in u2 around 0

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

      1. *-rgt-identity N/A

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

      2. sub-neg N/A

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

      3. rgt-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-rgt-identity N/A

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

      14. lower-/.f32 N/A

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

      15. associate-*r* N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   5. Applied rewrites 98.3%

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

Alternative 7: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.999986469745636:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.999986469745636)
   (* (fma u2 (* u2 -19.739208802181317) 1.0) (sqrt u1))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.999986469745636f) {
		tmp = fmaf(u2, (u2 * -19.739208802181317f), 1.0f) * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(6.28318530718) * u2)) <= Float32(0.999986469745636))
		tmp = Float32(fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0)) * sqrt(u1));
	else
		tmp = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99998647

   1. Initial program 97.6%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. lift-/.f32 N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. lower-pow.f32 N/A

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

      8. lift--.f32 N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

          \[\leadsto {\color{blue}{\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)}}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      11. *-inverses N/A

          \[\leadsto {\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)}^{\left(-1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. metadata-eval 97.7

          \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. Applied rewrites 97.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. sub-neg N/A

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

      11. remove-double-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f32 N/A

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

      17. sub-neg N/A

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

      18. distribute-lft-in N/A

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

      19. metadata-eval N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 54.0%

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

   if 0.99998647 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.6%

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

   3. Taylor expanded in u2 around 0

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

      1. *-rgt-identity N/A

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

      2. sub-neg N/A

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

      3. rgt-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-rgt-identity N/A

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

      14. lower-/.f32 N/A

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

      15. associate-*r* N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   5. Applied rewrites 98.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.999986469745636:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.7% 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 99.0%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 94.0%

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

Alternative 9: 93.6% 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 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} + \frac{-961389193575684075633145058384385882649239799132134631991269883031841}{11250000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
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 94.0

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

5. Applied rewrites 94.0%

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

Alternative 10: 91.7% 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 99.0%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(\frac{9740909103402808085817682884085781839780052161}{150000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{98696044010906577398881}{5000000000000000000000}\right)\right)} \]
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 91.3

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

5. Applied rewrites 91.3%

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

Alternative 11: 88.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 98.8%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. frac-sub N/A

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

   8. sub-div N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. cube-unmult N/A

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

   13. lift-+.f32 N/A

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. frac-times N/A

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

   7. *-lft-identity N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. lift--.f32 N/A

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

   11. metadata-eval N/A

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

   12. lift-*.f32 N/A

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

   13. lift-+.f32 N/A

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

   14. flip-- N/A

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

8. Applied rewrites 98.6%

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

9. Taylor expanded in u2 around 0

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

11. Applied rewrites 88.4%

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

12. Final simplification 88.4%

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

Alternative 12: 80.4% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 80.7%

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

Alternative 13: 74.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 80.7%

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

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 74.4%

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

Alternative 14: 71.7% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 80.7%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 70.9%

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

Alternative 15: 63.3% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 80.7%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 63.0%

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

Reproduce

herbie shell --seed 2024223 
(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))))