Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 9.0s

Alternatives: 12

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: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ t_1 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_1 \leq 0.9984999895095825:\\ \;\;\;\;\sqrt{u1 \cdot u1 + u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-19.739208802181317 \cdot u2\right) \cdot u2\right) \cdot t\_0 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))) (t_1 (cos (* 6.28318530718 u2))))
   (if (<= t_1 0.9984999895095825)
     (* (sqrt (+ (* u1 u1) u1)) t_1)
     (+ (* (* (* -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)));
	float t_1 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_1 <= 0.9984999895095825f) {
		tmp = sqrtf(((u1 * u1) + u1)) * t_1;
	} else {
		tmp = (((-19.739208802181317f * u2) * u2) * t_0) + t_0;
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = sqrt((u1 / (1.0e0 - u1)))
    t_1 = cos((6.28318530718e0 * u2))
    if (t_1 <= 0.9984999895095825e0) then
        tmp = sqrt(((u1 * u1) + u1)) * t_1
    else
        tmp = ((((-19.739208802181317e0) * u2) * u2) * t_0) + t_0
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	t_1 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.9984999895095825))
		tmp = Float32(sqrt(Float32(Float32(u1 * u1) + u1)) * t_1);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-19.739208802181317) * u2) * u2) * t_0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt((u1 / (single(1.0) - u1)));
	t_1 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_1 <= single(0.9984999895095825))
		tmp = sqrt(((u1 * u1) + u1)) * t_1;
	else
		tmp = (((single(-19.739208802181317) * u2) * u2) * t_0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99849999

   1. Initial program 98.0%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 8.9

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 87.6%

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

   if 0.99849999 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0

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

      1. 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 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f32 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

   8. Applied rewrites 92.7%

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

   10. Applied rewrites 99.3%

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

Alternative 3: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ t_1 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;t\_0 \leq 0.9984999895095825:\\ \;\;\;\;\sqrt{\left(u1 + 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-19.739208802181317 \cdot u2\right) \cdot u2\right) \cdot t\_1 + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))) (t_1 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= t_0 0.9984999895095825)
     (* (sqrt (* (+ u1 1.0) u1)) t_0)
     (+ (* (* (* -19.739208802181317 u2) u2) t_1) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float t_1 = sqrtf((u1 / (1.0f - u1)));
	float tmp;
	if (t_0 <= 0.9984999895095825f) {
		tmp = sqrtf(((u1 + 1.0f) * u1)) * t_0;
	} else {
		tmp = (((-19.739208802181317f * u2) * u2) * t_1) + t_1;
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = cos((6.28318530718e0 * u2))
    t_1 = sqrt((u1 / (1.0e0 - u1)))
    if (t_0 <= 0.9984999895095825e0) then
        tmp = sqrt(((u1 + 1.0e0) * u1)) * t_0
    else
        tmp = ((((-19.739208802181317e0) * u2) * u2) * t_1) + t_1
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	t_1 = sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9984999895095825))
		tmp = Float32(sqrt(Float32(Float32(u1 + Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-19.739208802181317) * u2) * u2) * t_1) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	t_1 = sqrt((u1 / (single(1.0) - u1)));
	tmp = single(0.0);
	if (t_0 <= single(0.9984999895095825))
		tmp = sqrt(((u1 + single(1.0)) * u1)) * t_0;
	else
		tmp = (((single(-19.739208802181317) * u2) * u2) * t_1) + t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2)) < 0.99849999

   1. Initial program 98.0%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 11.1

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

   5. Applied rewrites 8.3%

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

   7. Applied rewrites 87.4%

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

   if 0.99849999 < (cos.f32 (*.f32 #s(literal 314159265359/50000000000 binary32) u2))

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0

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

      1. 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 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f32 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

   8. Applied rewrites 92.7%

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

   10. Applied rewrites 99.3%

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

Alternative 4: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\left(\left(-19.739208802181317 \cdot u2\right) \cdot u2\right) \cdot t\_0 + t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ u1 (- 1.0 u1)))))
   (if (<= (* 6.28318530718 u2) 0.15000000596046448)
     (+ (* (* (* -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) <= 0.15000000596046448f) {
		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) <= 0.15000000596046448e0) 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(0.15000000596046448))
		tmp = Float32(Float32(Float32(Float32(Float32(-19.739208802181317) * 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(0.15000000596046448))
		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) < 0.150000006

   1. Initial program 99.4%

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

   3. Taylor expanded in u2 around 0

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

      1. 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 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. lower-/.f32 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

      19. fp-cancel-sign-sub-inv N/A

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

   8. Applied rewrites 90.6%

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

   10. Applied rewrites 98.5%

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

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

   1. Initial program 97.7%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 73.9

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

   5. Applied rewrites 73.9%

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

Alternative 5: 88.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 / (1.0f - u1)));
	return (((-19.739208802181317f * u2) * t_0) * u2) + 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) * t_0) * u2) + 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(Float32(-19.739208802181317) * u2) * t_0) * u2) + t_0)
end

MATLAB

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

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. 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 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. lower-/.f32 N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower--.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-lft-identity N/A

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

   18. metadata-eval N/A

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

   19. fp-cancel-sign-sub-inv N/A

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

8. Applied rewrites 80.1%

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

10. Applied rewrites 88.1%

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

12. Applied rewrites 88.1%

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

Alternative 6: 88.5% accurate, 2.0× speedup

Math

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

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(Float32(-19.739208802181317) * 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

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. 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 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. lower-/.f32 N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower--.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-lft-identity N/A

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

   18. metadata-eval N/A

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

   19. fp-cancel-sign-sub-inv N/A

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

8. Applied rewrites 79.8%

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

10. Applied rewrites 88.1%

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

Alternative 7: 88.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (sqrt(u1) + ((sqrt(u1) * single(-19.739208802181317)) * (u2 * u2))) / sqrt((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. 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 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. lower-/.f32 N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower--.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-lft-identity N/A

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

   18. metadata-eval N/A

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

   19. fp-cancel-sign-sub-inv N/A

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

8. Applied rewrites 79.8%

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

10. Applied rewrites 88.1%

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

12. Applied rewrites 87.7%

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

Alternative 8: 87.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(-19.739208802181317) * sqrt(u1)) * u2) * u2) + 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. 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 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. lower-/.f32 N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower--.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-sqrt.f32 N/A

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

   17. *-lft-identity N/A

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

   18. metadata-eval N/A

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

   19. fp-cancel-sign-sub-inv N/A

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

8. Applied rewrites 80.1%

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

10. Applied rewrites 88.1%

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

11. Taylor expanded in u1 around 0

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

13. Applied rewrites 87.7%

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

Alternative 9: 80.0% 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. 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 80.1

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

5. Applied rewrites 80.1%

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

Alternative 10: 71.7% accurate, 7.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 * u1) + 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 * u1) + u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(u1 * u1) + u1))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u2 around 0

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

   1. 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 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 64.1%

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

10. Applied rewrites 72.2%

    \[\leadsto \sqrt{u1 \cdot u1 + u1} \]
11. Add Preprocessing

Alternative 11: 71.6% accurate, 7.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (u1 - -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 = sqrt((u1 * (u1 - (-1.0e0))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (u1 - 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. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. 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 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 64.1%

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

10. Applied rewrites 72.0%

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

Alternative 12: 63.2% 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. 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 80.1

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

5. Applied rewrites 80.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 64.1%

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

Reproduce

herbie shell --seed 2024333 
(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))))