Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 8.6s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

Alternative 2: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.14000000059604645:\\ \;\;\;\;\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-1 - u1\right) \cdot \left(-u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.14000000059604645)
     (+ (* (* -19.739208802181317 (* u2 u2)) t_0) t_0)
     (* (sqrt (* (- -1.0 u1) (- u1))) (cos (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 0.14000000059604645f) {
		tmp = ((-19.739208802181317f * (u2 * u2)) * t_0) + t_0;
	} else {
		tmp = sqrtf(((-1.0f - u1) * -u1)) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    if ((6.28318530718e0 * u2) <= 0.14000000059604645e0) then
        tmp = (((-19.739208802181317e0) * (u2 * u2)) * t_0) + t_0
    else
        tmp = sqrt((((-1.0e0) - u1) * -u1)) * cos((6.28318530718e0 * u2))
    end if
    code = tmp
end function

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(0.14000000059604645))
		tmp = Float32(Float32(Float32(Float32(-19.739208802181317) * Float32(u2 * u2)) * t_0) + t_0);
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(-1.0) - u1) * Float32(-u1))) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.14000000059604645))
		tmp = ((single(-19.739208802181317) * (u2 * u2)) * t_0) + t_0;
	else
		tmp = sqrt(((single(-1.0) - u1) * -u1)) * cos((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.140000001

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. *-rgt-identity N/A

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

      10. sub-neg N/A

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

      11. rgt-mult-inverse N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. mul-1-neg N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-lft-in N/A

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. associate-*r* N/A

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

   5. Applied rewrites 91.4%

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

   7. Applied rewrites 98.7%

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

   if 0.140000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

      1. lift-/.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 97.4

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

   4. Applied rewrites 97.4%

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

   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 87.4

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

   7. Applied rewrites 87.4%

      \[\leadsto \sqrt{\color{blue}{\left(-1 - u1\right)} \cdot \left(-u1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.1995999962091446:\\ \;\;\;\;\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right)\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.1995999962091446)
     (+ (* (* -19.739208802181317 (* u2 u2)) t_0) t_0)
     (* (sqrt u1) (cos (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if ((6.28318530718f * u2) <= 0.1995999962091446f) {
		tmp = ((-19.739208802181317f * (u2 * u2)) * t_0) + t_0;
	} else {
		tmp = sqrtf(u1) * cosf((6.28318530718f * u2));
	}
	return tmp;
}

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    if ((6.28318530718e0 * u2) <= 0.1995999962091446e0) then
        tmp = (((-19.739208802181317e0) * (u2 * u2)) * t_0) + t_0
    else
        tmp = sqrt(u1) * cos((6.28318530718e0 * u2))
    end if
    code = tmp
end function

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(0.1995999962091446))
		tmp = Float32(Float32(Float32(Float32(-19.739208802181317) * Float32(u2 * u2)) * t_0) + t_0);
	else
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.1995999962091446))
		tmp = ((single(-19.739208802181317) * (u2 * u2)) * t_0) + t_0;
	else
		tmp = sqrt(u1) * cos((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.199599996

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. *-rgt-identity N/A

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

      10. sub-neg N/A

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

      11. rgt-mult-inverse N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. mul-1-neg N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-lft-in N/A

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. associate-*r* N/A

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 98.2%

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

   if 0.199599996 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   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 75.5

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

   5. Applied rewrites 75.5%

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

Alternative 4: 88.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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
    real(4) :: t_0
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    code = (((-19.739208802181317e0) * (u2 * u2)) * t_0) + t_0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	tmp = ((single(-19.739208802181317) * (u2 * u2)) * t_0) + t_0;
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. *-rgt-identity N/A

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

   10. sub-neg N/A

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

   11. rgt-mult-inverse N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. *-rgt-identity N/A

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

   16. distribute-lft-in N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. associate-*r* N/A

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

5. Applied rewrites 79.1%

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

7. Applied rewrites 87.6%

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

Alternative 5: 88.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return ((-19.739208802181317f * (u2 * u2)) + 1.0f) * 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 = (((-19.739208802181317e0) * (u2 * u2)) + 1.0e0) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(-19.739208802181317) * (u2 * u2)) + single(1.0)) * 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

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt1-in N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. *-rgt-identity N/A

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

   10. sub-neg N/A

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

   11. rgt-mult-inverse N/A

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

   12. mul-1-neg N/A

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

   13. distribute-neg-frac2 N/A

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

   14. mul-1-neg N/A

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

   15. *-rgt-identity N/A

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

   16. distribute-lft-in N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. associate-*r* N/A

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

5. Applied rewrites 79.5%

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

7. Applied rewrites 87.5%

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

Alternative 6: 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 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 79.7%

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

Alternative 7: 71.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(u1 + 1\right) \cdot u1} \end{array} \]

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(Float32(u1 + 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 79.7%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 63.1%

   \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 71.8%

    \[\leadsto \sqrt{\left(u1 + 1\right) \cdot u1} \]
11. Add Preprocessing

Alternative 8: 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 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 79.7%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 63.1%

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

Reproduce

herbie shell --seed 2024314 
(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))))