Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 8.5s

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 98.9%

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

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0015899999998509884:\\ \;\;\;\;\sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0015899999998509884)
   (sqrt (/ u1 (+ (- 2.0 u1) -1.0)))
   (* (cos (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0015899999998509884f) {
		tmp = sqrtf((u1 / ((2.0f - u1) + -1.0f)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	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) :: tmp
    if ((6.28318530718e0 * u2) <= 0.0015899999998509884e0) then
        tmp = sqrt((u1 / ((2.0e0 - u1) + (-1.0e0))))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0015899999998509884))
		tmp = sqrt(Float32(u1 / Float32(Float32(Float32(2.0) - u1) + Float32(-1.0))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0015899999998509884))
		tmp = sqrt((u1 / ((single(2.0) - u1) + single(-1.0))));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00159

   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 99.1%

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

      1. expm1-log1p-u 99.0%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \]

   5. Applied egg-rr 99.0%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \]
   6. Step-by-step derivation

      1. expm1-undefine 99.0%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} - 1}}} \]

      2. sub-neg 99.0%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} + \left(-1\right)}}} \]

      3. log1p-undefine 99.1%

         \[\leadsto \sqrt{\frac{u1}{e^{\color{blue}{\log \left(1 + \left(1 - u1\right)\right)}} + \left(-1\right)}} \]

      4. rem-exp-log 99.1%

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

      5. associate-+r- 99.2%

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

      6. metadata-eval 99.2%

         \[\leadsto \sqrt{\frac{u1}{\left(\color{blue}{2} - u1\right) + \left(-1\right)}} \]

      7. metadata-eval 99.2%

         \[\leadsto \sqrt{\frac{u1}{\left(2 - u1\right) + \color{blue}{-1}}} \]

   7. Simplified 99.2%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(2 - u1\right) + -1}}} \]

   if 0.00159 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0 89.4%

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

      1. +-commutative 89.4%

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

   5. Simplified 89.4%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0015899999998509884:\\ \;\;\;\;\sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004900000058114529:\\ \;\;\;\;\sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004900000058114529)
   (sqrt (/ u1 (+ (- 2.0 u1) -1.0)))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004900000058114529f) {
		tmp = sqrtf((u1 / ((2.0f - u1) + -1.0f)));
	} else {
		tmp = cosf((6.28318530718f * u2)) * sqrtf(u1);
	}
	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) :: tmp
    if ((6.28318530718e0 * u2) <= 0.004900000058114529e0) then
        tmp = sqrt((u1 / ((2.0e0 - u1) + (-1.0e0))))
    else
        tmp = cos((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.004900000058114529))
		tmp = sqrt(Float32(u1 / Float32(Float32(Float32(2.0) - u1) + Float32(-1.0))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.004900000058114529))
		tmp = sqrt((u1 / ((single(2.0) - u1) + single(-1.0))));
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00490000006

   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 98.1%

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

      1. expm1-log1p-u 97.9%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \]

   5. Applied egg-rr 97.9%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \]
   6. Step-by-step derivation

      1. expm1-undefine 98.0%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} - 1}}} \]

      2. sub-neg 98.0%

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} + \left(-1\right)}}} \]

      3. log1p-undefine 98.0%

         \[\leadsto \sqrt{\frac{u1}{e^{\color{blue}{\log \left(1 + \left(1 - u1\right)\right)}} + \left(-1\right)}} \]

      4. rem-exp-log 98.0%

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

      5. associate-+r- 98.1%

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

      6. metadata-eval 98.1%

         \[\leadsto \sqrt{\frac{u1}{\left(\color{blue}{2} - u1\right) + \left(-1\right)}} \]

      7. metadata-eval 98.1%

         \[\leadsto \sqrt{\frac{u1}{\left(2 - u1\right) + \color{blue}{-1}}} \]

   7. Simplified 98.1%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(2 - u1\right) + -1}}} \]

   if 0.00490000006 < (*.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 77.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004900000058114529:\\ \;\;\;\;\sqrt{\frac{u1}{\left(2 - u1\right) + -1}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{\left(2 - u1\right) + -1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (/ u1 (+ (- 2.0 u1) -1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / ((2.0f - 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 / ((2.0e0 - u1) + (-1.0e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 / Float32(Float32(Float32(2.0) - u1) + Float32(-1.0))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / ((single(2.0) - u1) + single(-1.0))));
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 81.9%

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

   1. expm1-log1p-u 81.8%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \]

5. Applied egg-rr 81.8%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - u1\right)\right)}}} \]
6. Step-by-step derivation

   1. expm1-undefine 81.8%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} - 1}}} \]

   2. sub-neg 81.8%

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{e^{\mathsf{log1p}\left(1 - u1\right)} + \left(-1\right)}}} \]

   3. log1p-undefine 81.9%

      \[\leadsto \sqrt{\frac{u1}{e^{\color{blue}{\log \left(1 + \left(1 - u1\right)\right)}} + \left(-1\right)}} \]

   4. rem-exp-log 81.9%

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

   5. associate-+r- 81.9%

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

   6. metadata-eval 81.9%

      \[\leadsto \sqrt{\frac{u1}{\left(\color{blue}{2} - u1\right) + \left(-1\right)}} \]

   7. metadata-eval 81.9%

      \[\leadsto \sqrt{\frac{u1}{\left(2 - u1\right) + \color{blue}{-1}}} \]

7. Simplified 81.9%

   \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(2 - u1\right) + -1}}} \]
8. Add Preprocessing

Alternative 5: 80.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 81.9%

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

Alternative 6: 71.7% accurate, 2.0× 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 98.9%

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

3. Taylor expanded in u2 around 0 81.9%

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

4. Taylor expanded in u1 around 0 73.2%

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

   1. +-commutative 87.4%

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

6. Simplified 73.2%

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

Alternative 7: 63.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {u1}^{0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (pow u1 0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(u1, 0.5f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u1 ** 0.5e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return u1 ^ Float32(0.5)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 ^ single(0.5);
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 81.9%

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

4. Taylor expanded in u1 around 0 64.8%

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

   1. pow1/2 64.8%

      \[\leadsto \color{blue}{{u1}^{0.5}} \]

6. Applied egg-rr 64.8%

   \[\leadsto \color{blue}{{u1}^{0.5}} \]
7. Add Preprocessing

Alternative 8: 63.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 81.9%

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

4. Taylor expanded in u1 around 0 64.8%

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

Reproduce

herbie shell --seed 2024185 
(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))))