Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 8.7s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 2: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t\_0 \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{\left(-u1\right) \cdot \left(-1 - u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right) \cdot u2\right) \cdot u2 + 1\right) \cdot \sqrt{\frac{-1}{u1 - 1} \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= (* t_0 (sqrt (/ u1 (- 1.0 u1)))) 0.02500000037252903)
     (* (sqrt (* (- u1) (- -1.0 u1))) t_0)
     (*
      (+
       (* (* (fma (* u2 u2) 64.93939402268539 -19.739208802181317) u2) u2)
       1.0)
      (sqrt (* (/ -1.0 (- u1 1.0)) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if ((t_0 * sqrtf((u1 / (1.0f - u1)))) <= 0.02500000037252903f) {
		tmp = sqrtf((-u1 * (-1.0f - u1))) * t_0;
	} else {
		tmp = (((fmaf((u2 * u2), 64.93939402268539f, -19.739208802181317f) * u2) * u2) + 1.0f) * sqrtf(((-1.0f / (u1 - 1.0f)) * u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (Float32(t_0 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))) <= Float32(0.02500000037252903))
		tmp = Float32(sqrt(Float32(Float32(-u1) * Float32(Float32(-1.0) - u1))) * t_0);
	else
		tmp = Float32(Float32(Float32(Float32(fma(Float32(u2 * u2), Float32(64.93939402268539), Float32(-19.739208802181317)) * u2) * u2) + Float32(1.0)) * sqrt(Float32(Float32(Float32(-1.0) / Float32(u1 - Float32(1.0))) * u1)));
	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))) < 0.0250000004

   1. Initial program 98.9%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 N/A

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

      9. lower-neg.f32 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in u1 around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 98.1

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

   7. Applied rewrites 98.1%

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

   if 0.0250000004 < (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))) (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)))

   1. Initial program 99.5%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 N/A

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

      9. lower-neg.f32 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 86.0

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

   7. Applied rewrites 86.0%

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

   9. Applied rewrites 93.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}} \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{\left(-u1\right) \cdot \left(-1 - u1\right)} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right) \cdot u2\right) \cdot u2 + 1\right) \cdot \sqrt{\frac{-1}{u1 - 1} \cdot u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.15000000596046448:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right) \cdot u2\right) \cdot u2 + 1\right) \cdot \sqrt{\frac{-1}{u1 - 1} \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.15000000596046448)
   (*
    (+ (* (* (fma (* u2 u2) 64.93939402268539 -19.739208802181317) u2) u2) 1.0)
    (sqrt (* (/ -1.0 (- u1 1.0)) u1)))
   (* (sqrt u1) (cos (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.15000000596046448f) {
		tmp = (((fmaf((u2 * u2), 64.93939402268539f, -19.739208802181317f) * u2) * u2) + 1.0f) * sqrtf(((-1.0f / (u1 - 1.0f)) * u1));
	} else {
		tmp = sqrtf(u1) * cosf((u2 * 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.15000000596046448))
		tmp = Float32(Float32(Float32(Float32(fma(Float32(u2 * u2), Float32(64.93939402268539), Float32(-19.739208802181317)) * u2) * u2) + Float32(1.0)) * sqrt(Float32(Float32(Float32(-1.0) / Float32(u1 - Float32(1.0))) * u1)));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.150000006

   1. Initial program 99.5%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 N/A

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

      9. lower-neg.f32 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 91.6

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

   7. Applied rewrites 91.6%

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

   9. Applied rewrites 98.4%

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

   if 0.150000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.2%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 77.3

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

   5. Applied rewrites 77.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.15000000596046448:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(u2 \cdot u2, 64.93939402268539, -19.739208802181317\right) \cdot u2\right) \cdot u2 + 1\right) \cdot \sqrt{\frac{-1}{u1 - 1} \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f32 N/A

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

   9. lower-neg.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 78.6

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

7. Applied rewrites 78.6%

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

9. Applied rewrites 86.8%

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

10. Final simplification 86.8%

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

Alternative 5: 79.9% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 78.7%

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

Alternative 6: 63.0% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-rgt-identity N/A

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

   2. sub-neg N/A

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

   3. rgt-mult-inverse N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. associate-*r* N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-rgt-identity N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

5. Applied rewrites 78.7%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 60.9%

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

Alternative 7: 4.1% accurate, 135.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. lift--.f32 N/A

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

   2. flip3-- N/A

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

   3. sqr-pow N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{{1}^{3} - \color{blue}{{u1}^{\left(\frac{3}{2}\right)} \cdot {u1}^{\left(\frac{3}{2}\right)}}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. pow-prod-down N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{{1}^{3} - \color{blue}{{\left(u1 \cdot u1\right)}^{\left(\frac{3}{2}\right)}}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. sqr-neg N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{{1}^{3} - {\color{blue}{\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. unpow-prod-down N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{{1}^{3} - \color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u1\right)\right)}^{\left(\frac{3}{2}\right)}}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. sqr-pow N/A

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

   8. flip3-+ N/A

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

   9. *-lft-identity N/A

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

   10. sqr-pow N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{{1}^{3} - {\left(\mathsf{neg}\left(u1\right)\right)}^{3}}{1 \cdot 1 + \frac{{\left(u1 \cdot u1\right)}^{3} + \color{blue}{{u1}^{\left(\frac{3}{2}\right)} \cdot {u1}^{\left(\frac{3}{2}\right)}}}{\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right) + \left(\left(1 \cdot u1\right) \cdot \left(1 \cdot u1\right) - \left(u1 \cdot u1\right) \cdot \left(1 \cdot u1\right)\right)}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. pow-prod-down N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{{1}^{3} - {\left(\mathsf{neg}\left(u1\right)\right)}^{3}}{1 \cdot 1 + \frac{{\left(u1 \cdot u1\right)}^{3} + \color{blue}{{\left(u1 \cdot u1\right)}^{\left(\frac{3}{2}\right)}}}{\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right) + \left(\left(1 \cdot u1\right) \cdot \left(1 \cdot u1\right) - \left(u1 \cdot u1\right) \cdot \left(1 \cdot u1\right)\right)}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. sqr-neg N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{{1}^{3} - {\left(\mathsf{neg}\left(u1\right)\right)}^{3}}{1 \cdot 1 + \frac{{\left(u1 \cdot u1\right)}^{3} + {\color{blue}{\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right) + \left(\left(1 \cdot u1\right) \cdot \left(1 \cdot u1\right) - \left(u1 \cdot u1\right) \cdot \left(1 \cdot u1\right)\right)}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. unpow-prod-down N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{{1}^{3} - {\left(\mathsf{neg}\left(u1\right)\right)}^{3}}{1 \cdot 1 + \frac{{\left(u1 \cdot u1\right)}^{3} + \color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u1\right)\right)}^{\left(\frac{3}{2}\right)}}}{\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right) + \left(\left(1 \cdot u1\right) \cdot \left(1 \cdot u1\right) - \left(u1 \cdot u1\right) \cdot \left(1 \cdot u1\right)\right)}}}} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   14. sqr-pow N/A

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

   15. *-lft-identity N/A

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

   16. *-lft-identity N/A

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

   17. sqr-neg N/A

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

   18. *-lft-identity N/A

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

4. Applied rewrites 71.5%

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

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-+.f32 59.1

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

7. Applied rewrites 59.1%

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

8. Taylor expanded in u1 around -inf

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

10. Applied rewrites 4.4%

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

Reproduce

herbie shell --seed 2024285 
(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))))