Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 15.0s

Alternatives: 23

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 98.9%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

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) 0.05000000074505806)
     (* t_1 (sqrt (fma u1 (fma u1 u1 u1) u1)))
     (+
      t_0
      (*
       t_0
       (*
        (* u2 u2)
        (fma
         (* u2 u2)
         (fma (* u2 u2) -85.45681720672748 64.93939402268539)
         -19.739208802181317)))))))

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) <= 0.05000000074505806f) {
		tmp = t_1 * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = t_0 + (t_0 * ((u2 * u2) * fmaf((u2 * u2), fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f), -19.739208802181317f)));
	}
	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(0.05000000074505806))
		tmp = Float32(t_1 * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = Float32(t_0 + Float32(t_0 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)), Float32(-19.739208802181317)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.7%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(1 + u1\right)\right)}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   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. accelerator-lowering-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. accelerator-lowering-fma.f32 97.5

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

   5. Simplified 97.5%

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

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

   1. Initial program 99.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. Simplified 98.0%

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

      1. +-lowering-+.f32 N/A

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

   7. Applied egg-rr 98.0%

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

4. Final simplification 97.6%

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

Alternative 3: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.01899999938905239:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.01899999938905239)
     (*
      (sqrt (fma u1 u1 u1))
      (fma
       (* u2 u2)
       (fma
        (* u2 u2)
        (fma (* u2 u2) -85.45681720672748 64.93939402268539)
        -19.739208802181317)
       1.0))
     (* t_0 (fma u2 (* u2 -19.739208802181317) 1.0)))))

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))) <= 0.01899999938905239f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * fmaf((u2 * u2), fmaf((u2 * u2), fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f), -19.739208802181317f), 1.0f);
	} else {
		tmp = t_0 * fmaf(u2, (u2 * -19.739208802181317f), 1.0f);
	}
	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(0.01899999938905239))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)), Float32(-19.739208802181317)), Float32(1.0)));
	else
		tmp = Float32(t_0 * fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))) < 0.0189999994

   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. accelerator-lowering-fma.f32 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f32 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f32 90.9

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

   8. Simplified 90.9%

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

   if 0.0189999994 < (*.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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f32 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. *-rgt-identity N/A

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

      11. sub-neg N/A

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

      12. rgt-mult-inverse N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. distribute-lft-in N/A

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

      18. +-commutative N/A

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

      19. sub-neg N/A

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

      20. associate-*r* N/A

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

   5. Simplified 93.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.01899999938905239:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.13500000536441803:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.13500000536441803)
     (*
      (sqrt (fma u1 (fma u1 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))) <= 0.13500000536441803f) {
		tmp = sqrtf(fmaf(u1, fmaf(u1, 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(0.13500000536441803))
		tmp = Float32(sqrt(fma(u1, fma(u1, 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))) < 0.135000005

   1. Initial program 98.8%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f32 N/A

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

      7. cos-lowering-cos.f32 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. sqrt-lowering-sqrt.f32 N/A

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

      10. div-sub N/A

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

      11. sub-neg N/A

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

      12. *-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f32 N/A

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

      15. /-lowering-/.f32 98.5

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

   4. Applied egg-rr 98.5%

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

   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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.f32 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. +-lowering-+.f32 N/A

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

      15. /-lowering-/.f32 86.4

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

   7. Simplified 86.4%

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

   8. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. accelerator-lowering-fma.f32 86.2

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

   10. Simplified 86.2%

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

   if 0.135000005 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. sqrt-lowering-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. /-lowering-/.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. Simplified 89.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.13500000536441803:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* t_0 (cos (* 6.28318530718 u2))) 0.02199999988079071)
     (* (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))) <= 0.02199999988079071f) {
		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(0.02199999988079071))
		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))) < 0.0219999999

   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. accelerator-lowering-fma.f32 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-lowering-*.f32 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. unpow2 N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-in N/A

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

      12. sqrt-lowering-sqrt.f32 N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. accelerator-lowering-fma.f32 86.0

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

   8. Simplified 86.0%

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

   if 0.0219999999 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

      1. *-rgt-identity N/A

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

      2. sub-neg N/A

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

      3. rgt-mult-inverse N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. sqrt-lowering-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. /-lowering-/.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. Simplified 87.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.25:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot 64.93939402268539, \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.25)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (fma
     (* u2 u2)
     (* (* u2 u2) 64.93939402268539)
     (fma u2 (* u2 -19.739208802181317) 1.0)))
   (* (cos (* 6.28318530718 u2)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.25f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * fmaf((u2 * u2), ((u2 * u2) * 64.93939402268539f), fmaf(u2, (u2 * -19.739208802181317f), 1.0f));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 + (u1 * u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.25))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(u2 * u2), Float32(Float32(u2 * u2) * Float32(64.93939402268539)), fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.25

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

   5. Simplified 99.3%

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

   if 0.25 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. accelerator-lowering-fma.f32 83.2

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

   5. Simplified 83.2%

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

      1. +-lowering-+.f32 N/A

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

      2. *-lowering-*.f32 83.2

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

   7. Applied egg-rr 83.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.25:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot 64.93939402268539, \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.25:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot 64.93939402268539, \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.25)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (fma
     (* u2 u2)
     (* (* u2 u2) 64.93939402268539)
     (fma u2 (* u2 -19.739208802181317) 1.0)))
   (* (cos (* 6.28318530718 u2)) (sqrt (fma u1 u1 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.25f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * fmaf((u2 * u2), ((u2 * u2) * 64.93939402268539f), fmaf(u2, (u2 * -19.739208802181317f), 1.0f));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(fmaf(u1, u1, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.25))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * fma(Float32(u2 * u2), Float32(Float32(u2 * u2) * Float32(64.93939402268539)), fma(u2, Float32(u2 * Float32(-19.739208802181317)), Float32(1.0))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.25

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

   5. Simplified 99.3%

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

   if 0.25 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. accelerator-lowering-fma.f32 83.2

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

   5. Simplified 83.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.25:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(u2 \cdot u2\right) \cdot 64.93939402268539, \mathsf{fma}\left(u2, u2 \cdot -19.739208802181317, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.999970018863678:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* 6.28318530718 u2)) 0.999970018863678)
   (* (sqrt u1) (fma -19.739208802181317 (* u2 u2) 1.0))
   (sqrt (/ u1 (- 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((6.28318530718f * u2)) <= 0.999970018863678f) {
		tmp = sqrtf(u1) * fmaf(-19.739208802181317f, (u2 * u2), 1.0f);
	} 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(0.999970018863678))
		tmp = Float32(sqrt(u1) * fma(Float32(-19.739208802181317), Float32(u2 * u2), Float32(1.0)));
	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)) < 0.999970019

   1. Initial program 97.8%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f32 N/A

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

      7. cos-lowering-cos.f32 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. sqrt-lowering-sqrt.f32 N/A

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

      10. div-sub N/A

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

      11. sub-neg N/A

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

      12. *-inverses N/A

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

      13. metadata-eval N/A

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

      14. +-lowering-+.f32 N/A

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

      15. /-lowering-/.f32 97.9

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

   4. Applied egg-rr 97.9%

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

   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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.f32 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. +-lowering-+.f32 N/A

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

      15. /-lowering-/.f32 64.5

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

   7. Simplified 64.5%

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

   8. Taylor expanded in u1 around 0

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

      1. sqrt-lowering-sqrt.f32 57.4

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

   10. Simplified 57.4%

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

   if 0.999970019 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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. sqrt-lowering-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. /-lowering-/.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. Simplified 97.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.999970018863678:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(-19.739208802181317, u2 \cdot u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.800000011920929:\\ \;\;\;\;t\_0 + t\_0 \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.800000011920929)
     (+
      t_0
      (*
       t_0
       (*
        (* u2 u2)
        (fma
         (* u2 u2)
         (fma (* u2 u2) -85.45681720672748 64.93939402268539)
         -19.739208802181317))))
     (* (cos (* 6.28318530718 u2)) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 0.800000011920929f) {
		tmp = t_0 + (t_0 * ((u2 * u2) * fmaf((u2 * u2), fmaf((u2 * u2), -85.45681720672748f, 64.93939402268539f), -19.739208802181317f)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.800000011920929))
		tmp = Float32(t_0 + Float32(t_0 * Float32(Float32(u2 * u2) * fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)), Float32(-19.739208802181317)))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.800000012

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0

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

   5. Simplified 99.0%

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

      1. +-lowering-+.f32 N/A

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

   7. Applied egg-rr 99.1%

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

   if 0.800000012 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 95.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sqrt-lowering-sqrt.f32 N/A

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

      3. cos-lowering-cos.f32 N/A

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

      4. *-lowering-*.f32 68.9

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

   5. Simplified 68.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.800000011920929:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} + \sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(u2 \cdot u2\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, -85.45681720672748, 64.93939402268539\right), -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 92.4%

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

   1. +-lowering-+.f32 N/A

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

7. Applied egg-rr 92.5%

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

8. Final simplification 92.5%

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

Alternative 11: 93.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 92.4%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

7. Applied egg-rr 92.5%

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

8. Final simplification 92.5%

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

Alternative 12: 93.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 92.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

7. Applied egg-rr 92.5%

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

Alternative 13: 93.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 92.4%

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

   1. *-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

7. Applied egg-rr 92.4%

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

Alternative 14: 93.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

   7. /-lowering-/.f32 N/A

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

   8. neg-lowering-neg.f32 N/A

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

   9. *-lft-identity N/A

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

   10. accelerator-lowering-fma.f32 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

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

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

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

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

   17. sqr-neg N/A

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

   18. +-lowering-+.f32 N/A

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

   19. *-lowering-*.f32 98.8

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. sub-neg N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. accelerator-lowering-fma.f32 N/A

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

   10. *-lowering-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f32 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f32 92.4

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

7. Simplified 92.4%

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

8. Final simplification 92.4%

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

Alternative 15: 93.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* u1 (/ -1.0 (+ u1 -1.0))))
  (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 + -1.0f)))) * 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) / Float32(u1 + Float32(-1.0))))) * fma(Float32(u2 * u2), fma(Float32(u2 * u2), fma(Float32(u2 * u2), Float32(-85.45681720672748), Float32(64.93939402268539)), Float32(-19.739208802181317)), Float32(1.0)))
end

Derivation

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. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

   7. /-lowering-/.f32 N/A

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

   8. neg-lowering-neg.f32 N/A

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

   9. *-lft-identity N/A

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

   10. accelerator-lowering-fma.f32 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

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

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

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

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

   17. sqr-neg N/A

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

   18. +-lowering-+.f32 N/A

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

   19. *-lowering-*.f32 98.8

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

4. Applied egg-rr 98.8%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

   3. remove-double-neg N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-/l/ N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. flip-- N/A

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

   13. associate-/r/ N/A

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

   14. *-lowering-*.f32 N/A

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f32 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. accelerator-lowering-fma.f32 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f32 92.3

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

9. Simplified 92.3%

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

10. Final simplification 92.3%

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

Alternative 16: 91.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 90.5%

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

Alternative 17: 91.6% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* u1 (/ -1.0 (+ u1 -1.0))))
  (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 + -1.0f)))) * fmaf((u2 * u2), fmaf((u2 * u2), 64.93939402268539f, -19.739208802181317f), 1.0f);
}

Julia

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

Derivation

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. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. frac-2neg N/A

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

   7. /-lowering-/.f32 N/A

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

   8. neg-lowering-neg.f32 N/A

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

   9. *-lft-identity N/A

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

   10. accelerator-lowering-fma.f32 N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

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

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

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

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

   17. sqr-neg N/A

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

   18. +-lowering-+.f32 N/A

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

   19. *-lowering-*.f32 98.8

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

4. Applied egg-rr 98.8%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

   3. remove-double-neg N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-/l/ N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. flip-- N/A

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

   13. associate-/r/ N/A

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

   14. *-lowering-*.f32 N/A

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 90.3

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

9. Simplified 90.3%

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

10. Final simplification 90.3%

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

Alternative 18: 88.5% 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 98.9%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. accelerator-lowering-fma.f32 N/A

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

   9. *-lowering-*.f32 N/A

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

   10. *-rgt-identity N/A

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

   11. sub-neg N/A

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

   12. rgt-mult-inverse N/A

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

   13. mul-1-neg N/A

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

   14. distribute-neg-frac2 N/A

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

   15. mul-1-neg N/A

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

   16. *-rgt-identity N/A

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

   17. distribute-lft-in N/A

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

   18. +-commutative N/A

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

   19. sub-neg N/A

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

   20. associate-*r* N/A

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

5. Simplified 88.0%

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

6. Final simplification 88.0%

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

Alternative 19: 80.2% 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 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
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. sqrt-lowering-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. /-lowering-/.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. Simplified 78.5%

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

Alternative 20: 74.7% 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 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
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. sqrt-lowering-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. /-lowering-/.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. Simplified 78.5%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. accelerator-lowering-fma.f32 72.9

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

8. Simplified 72.9%

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

Alternative 21: 71.9% 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 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
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. sqrt-lowering-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. /-lowering-/.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. Simplified 78.5%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. accelerator-lowering-fma.f32 70.2

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

8. Simplified 70.2%

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

Alternative 22: 63.4% 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 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
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. sqrt-lowering-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. /-lowering-/.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. Simplified 78.5%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 62.2

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

8. Simplified 62.2%

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

Alternative 23: 19.2% accurate, 135.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
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. sqrt-lowering-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. /-lowering-/.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. Simplified 78.5%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. accelerator-lowering-fma.f32 70.2

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

8. Simplified 70.2%

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

9. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{u1} \]
10. Step-by-step derivation

11. Simplified 18.5%

    \[\leadsto \color{blue}{u1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))