Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 12.1s

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 99.0%

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

3. Final simplification 99.0%

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

Alternative 2: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999988079071045:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9999988079071045)
     (* t_0 (sqrt (* u1 (+ u1 1.0))))
     (pow (+ (/ 1.0 u1) -1.0) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9999988079071045f) {
		tmp = t_0 * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f);
	}
	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 = cos((6.28318530718e0 * u2))
    if (t_0 <= 0.9999988079071045e0) then
        tmp = t_0 * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999988079071045))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((single(6.28318530718) * u2));
	tmp = single(0.0);
	if (t_0 <= single(0.9999988079071045))
		tmp = t_0 * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999998808

   1. Initial program 98.5%

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

      1. clear-num 98.2%

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

      2. associate-/r/ 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u1 around 0 86.1%

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

      1. +-commutative 86.1%

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

   7. Simplified 86.1%

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

   if 0.999998808 < (cos.f32 (*.f32 314159265359/50000000000 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 99.0%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. sqrt-pow1 99.1%

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

      2. div-sub 99.0%

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

      3. *-inverses 99.0%

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

      4. sub-neg 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

         \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9999988079071045:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.008999999612569809:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.008999999612569809)
   (pow (+ (/ 1.0 u1) -1.0) -0.5)
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.008999999612569809f) {
		tmp = powf(((1.0f / u1) + -1.0f), -0.5f);
	} 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) <= 0.008999999612569809e0) then
        tmp = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
    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(0.008999999612569809))
		tmp = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5);
	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(0.008999999612569809))
		tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00899999961

   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 96.8%

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

      1. clear-num 96.6%

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

      2. inv-pow 96.6%

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

   5. Applied egg-rr 96.6%

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

      1. sqrt-pow1 96.8%

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

      2. div-sub 96.8%

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

      3. *-inverses 96.8%

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

      4. sub-neg 96.8%

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

      5. metadata-eval 96.8%

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

      6. metadata-eval 96.8%

         \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]

   if 0.00899999961 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.3%

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

   3. Taylor expanded in u1 around 0 73.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.008999999612569809:\\ \;\;\;\;{\left(\frac{1}{u1} + -1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{u1} + -1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (pow (+ (/ 1.0 u1) -1.0) -0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f / u1) + -1.0f), -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 = ((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((single(1.0) / u1) + single(-1.0)) ^ single(-0.5);
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0 77.5%

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

   1. clear-num 77.4%

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

   2. inv-pow 77.4%

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

5. Applied egg-rr 77.4%

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

   1. sqrt-pow1 77.5%

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

   2. div-sub 77.5%

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

   3. *-inverses 77.5%

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

   4. sub-neg 77.5%

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

   5. metadata-eval 77.5%

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

   6. metadata-eval 77.5%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \]

7. Applied egg-rr 77.5%

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]

8. Final simplification 77.5%

   \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \]
9. Add Preprocessing

Alternative 5: 71.5% 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 99.0%

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.9%

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in u1 around 0 87.6%

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

   1. +-commutative 87.6%

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

7. Simplified 87.6%

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

8. Taylor expanded in u2 around 0 70.9%

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

   1. +-commutative 70.9%

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

10. Simplified 70.9%

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

11. Final simplification 70.9%

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

Alternative 6: 80.0% 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 99.0%

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

3. Taylor expanded in u2 around 0 77.5%

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

4. Final simplification 77.5%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
5. Add Preprocessing

Alternative 7: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{u1}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (pow (/ 1.0 u1) -0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((1.0f / 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 = (1.0e0 / u1) ** (-0.5e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) / u1) ^ Float32(-0.5)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(1.0) / u1) ^ single(-0.5);
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u2 around 0 77.5%

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

   1. clear-num 77.4%

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

   2. inv-pow 77.4%

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

5. Applied egg-rr 77.4%

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

   1. sqrt-pow1 77.5%

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

   2. div-sub 77.5%

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

   3. *-inverses 77.5%

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

   4. sub-neg 77.5%

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

   5. metadata-eval 77.5%

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

   6. metadata-eval 77.5%

      \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}} \]

7. Applied egg-rr 77.5%

   \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5}} \]

8. Taylor expanded in u1 around 0 63.3%

   \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \]

9. Final simplification 63.3%

   \[\leadsto {\left(\frac{1}{u1}\right)}^{-0.5} \]
10. Add Preprocessing

Alternative 8: 63.2% 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 99.0%

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

3. Taylor expanded in u2 around 0 77.5%

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

4. Taylor expanded in u1 around 0 63.3%

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

5. Final simplification 63.3%

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

Reproduce

herbie shell --seed 2024044 
(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))))