Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 18.2s

Alternatives: 10

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (sqrt (* (* u2 u2) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf(sqrtf(((u2 * u2) * 39.47841760436263f)));
}

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(sqrt(((u2 * u2) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(sqrt(Float32(Float32(u2 * u2) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos(sqrt(((u2 * u2) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}} \cdot \sqrt{6.28318530718 \cdot u2}\right) \]

   3. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(6.28318530718 \cdot u2\right)}^{0.5} \cdot \color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}}\right) \]

   4. pow-prod-down 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)\right)}^{0.5}\right)} \]

   5. swap-sqr 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\color{blue}{\left(\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)\right)}}^{0.5}\right) \]

   6. metadata-eval 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

   2. unpow2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{39.47841760436263 \cdot \color{blue}{{u2}^{2}}}\right) \]

   3. *-commutative 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}\right) \]

   4. unpow2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right)} \cdot 39.47841760436263}\right) \]

5. Simplified 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right)} \]

6. Final simplification 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]

Alternative 2: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (sqrt (* u2 (* u2 39.47841760436263))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf(sqrtf((u2 * (u2 * 39.47841760436263f))));
}

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(sqrt((u2 * (u2 * 39.47841760436263e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(sqrt(Float32(u2 * Float32(u2 * Float32(39.47841760436263))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos(sqrt((u2 * (u2 * single(39.47841760436263)))));
end

Derivation

1. Initial program 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}} \cdot \sqrt{6.28318530718 \cdot u2}\right) \]

   3. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(6.28318530718 \cdot u2\right)}^{0.5} \cdot \color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}}\right) \]

   4. pow-prod-down 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)\right)}^{0.5}\right)} \]

   5. swap-sqr 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\color{blue}{\left(\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)\right)}}^{0.5}\right) \]

   6. metadata-eval 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

   2. associate-*r* 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(39.47841760436263 \cdot u2\right) \cdot u2}}\right) \]

   3. *-commutative 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{u2 \cdot \left(39.47841760436263 \cdot u2\right)}}\right) \]

   4. *-commutative 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}}\right) \]

5. Simplified 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right)} \]

6. Final simplification 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

Alternative 3: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot 6.28318530718\right)\\ \mathbf{if}\;t_0 \leq 0.9764999747276306:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 6.28318530718))))
   (if (<= t_0 0.9764999747276306)
     (* t_0 (sqrt u1))
     (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* u2 (* u2 -19.739208802181317)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * 6.28318530718f));
	float tmp;
	if (t_0 <= 0.9764999747276306f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
	}
	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((u2 * 6.28318530718e0))
    if (t_0 <= 0.9764999747276306e0) then
        tmp = t_0 * sqrt(u1)
    else
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(6.28318530718)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9764999747276306))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = cos((u2 * single(6.28318530718)));
	tmp = single(0.0);
	if (t_0 <= single(0.9764999747276306))
		tmp = t_0 * sqrt(u1);
	else
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + (u2 * (u2 * single(-19.739208802181317))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.976499975

   1. Initial program 97.3%

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

   2. Taylor expanded in u1 around 0 73.3%

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

   if 0.976499975 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow1/2 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}} \cdot \sqrt{6.28318530718 \cdot u2}\right) \]

      3. pow1/2 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(6.28318530718 \cdot u2\right)}^{0.5} \cdot \color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}}\right) \]

      4. pow-prod-down 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)\right)}^{0.5}\right)} \]

      5. swap-sqr 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\color{blue}{\left(\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)\right)}}^{0.5}\right) \]

      6. metadata-eval 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

   3. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
   4. Step-by-step derivation

      1. unpow1/2 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

      2. associate-*r* 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(39.47841760436263 \cdot u2\right) \cdot u2}}\right) \]

      3. *-commutative 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{u2 \cdot \left(39.47841760436263 \cdot u2\right)}}\right) \]

      4. *-commutative 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}}\right) \]

   5. Simplified 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right)} \]

   6. Taylor expanded in u2 around 0 97.8%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
   7. Step-by-step derivation

      1. associate-*r* 97.8%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + \sqrt{\frac{u1}{1 - u1}} \]

      2. distribute-lft1-in 97.7%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. unpow2 97.7%

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

      4. associate-*r* 97.7%

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

   8. Simplified 97.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.9764999747276306:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \end{array} \]

Alternative 4: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.999750018119812:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* u2 6.28318530718)) 0.999750018119812)
   (* (sqrt u1) (+ 1.0 (* u2 (* u2 -19.739208802181317))))
   (pow (+ -1.0 (/ 1.0 u1)) -0.5)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((u2 * 6.28318530718f)) <= 0.999750018119812f) {
		tmp = sqrtf(u1) * (1.0f + (u2 * (u2 * -19.739208802181317f)));
	} else {
		tmp = powf((-1.0f + (1.0f / u1)), -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) :: tmp
    if (cos((u2 * 6.28318530718e0)) <= 0.999750018119812e0) then
        tmp = sqrt(u1) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
    else
        tmp = ((-1.0e0) + (1.0e0 / u1)) ** (-0.5e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(u2 * Float32(6.28318530718))) <= Float32(0.999750018119812))
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(u2 * Float32(u2 * Float32(-19.739208802181317)))));
	else
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) / u1)) ^ Float32(-0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (cos((u2 * single(6.28318530718))) <= single(0.999750018119812))
		tmp = sqrt(u1) * (single(1.0) + (u2 * (u2 * single(-19.739208802181317))));
	else
		tmp = (single(-1.0) + (single(1.0) / u1)) ^ single(-0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 314159265359/50000000000 u2)) < 0.999750018

   1. Initial program 97.8%

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

   2. Taylor expanded in u1 around 0 71.5%

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

   3. Taylor expanded in u2 around 0 49.0%

      \[\leadsto \color{blue}{\sqrt{u1} + -19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{u1}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 49.0%

         \[\leadsto \sqrt{u1} + \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{u1}} \]

      2. distribute-rgt1-in 49.0%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{u1}} \]

      3. unpow2 49.0%

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

      4. associate-*r* 49.0%

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

   5. Simplified 49.0%

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

   if 0.999750018 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow1/2 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}} \cdot \sqrt{6.28318530718 \cdot u2}\right) \]

      3. pow1/2 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(6.28318530718 \cdot u2\right)}^{0.5} \cdot \color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}}\right) \]

      4. pow-prod-down 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)\right)}^{0.5}\right)} \]

      5. swap-sqr 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\color{blue}{\left(\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)\right)}}^{0.5}\right) \]

      6. metadata-eval 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

   3. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
   4. Step-by-step derivation

      1. unpow1/2 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

      2. associate-*r* 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(39.47841760436263 \cdot u2\right) \cdot u2}}\right) \]

      3. *-commutative 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{u2 \cdot \left(39.47841760436263 \cdot u2\right)}}\right) \]

      4. *-commutative 99.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}}\right) \]

   5. Simplified 99.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right)} \]
   6. Step-by-step derivation

      1. clear-num 99.2%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      2. inv-pow 99.2%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{-1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      3. div-sub 99.2%

         \[\leadsto \sqrt{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      4. pow1 99.2%

         \[\leadsto \sqrt{{\left(\frac{1}{u1} - \frac{\color{blue}{{u1}^{1}}}{u1}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      5. pow1 99.2%

         \[\leadsto \sqrt{{\left(\frac{1}{u1} - \frac{{u1}^{1}}{\color{blue}{{u1}^{1}}}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      6. pow-div 99.2%

         \[\leadsto \sqrt{{\left(\frac{1}{u1} - \color{blue}{{u1}^{\left(1 - 1\right)}}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      7. metadata-eval 99.2%

         \[\leadsto \sqrt{{\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      8. metadata-eval 99.2%

         \[\leadsto \sqrt{{\left(\frac{1}{u1} - \color{blue}{1}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   7. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\color{blue}{{\left(\frac{1}{u1} - 1\right)}^{-1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]
   8. Step-by-step derivation

      1. unpow-1 99.2%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} - 1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      2. sub-neg 99.2%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

      3. metadata-eval 99.2%

         \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   9. Simplified 99.2%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} + -1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   10. Taylor expanded in u2 around 0 93.6%

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

       1. sub-neg 93.6%

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

       2. metadata-eval 93.6%

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

       3. unpow-1 93.6%

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

       4. metadata-eval 93.6%

          \[\leadsto \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       5. pow-sqr 93.8%

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

       6. rem-sqrt-square 93.8%

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

       7. sqr-pow 93.1%

          \[\leadsto \left|\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

       8. fabs-sqr 93.1%

          \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}} \]

       9. sqr-pow 93.8%

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

       10. +-commutative 93.8%

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

   12. Simplified 93.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot 6.28318530718\right) \leq 0.999750018119812:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + u2 \cdot \left(u2 \cdot -19.739208802181317\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \frac{1}{u1}\right)}^{-0.5}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((u2 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

2. Final simplification 98.8%

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

Alternative 6: 88.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

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))) * (1.0e0 + (u2 * (u2 * (-19.739208802181317e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}} \cdot \sqrt{6.28318530718 \cdot u2}\right) \]

   3. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(6.28318530718 \cdot u2\right)}^{0.5} \cdot \color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}}\right) \]

   4. pow-prod-down 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)\right)}^{0.5}\right)} \]

   5. swap-sqr 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\color{blue}{\left(\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)\right)}}^{0.5}\right) \]

   6. metadata-eval 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

   2. associate-*r* 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(39.47841760436263 \cdot u2\right) \cdot u2}}\right) \]

   3. *-commutative 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{u2 \cdot \left(39.47841760436263 \cdot u2\right)}}\right) \]

   4. *-commutative 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}}\right) \]

5. Simplified 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right)} \]

6. Taylor expanded in u2 around 0 87.3%

   \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \sqrt{\frac{u1}{1 - u1}}} \]
7. Step-by-step derivation

   1. associate-*r* 87.3%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + \sqrt{\frac{u1}{1 - u1}} \]

   2. distribute-lft1-in 87.2%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. unpow2 87.2%

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

   4. associate-*r* 87.2%

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

8. Simplified 87.2%

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

9. Final simplification 87.2%

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

Alternative 7: 79.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}} \cdot \sqrt{6.28318530718 \cdot u2}\right) \]

   3. pow1/2 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(6.28318530718 \cdot u2\right)}^{0.5} \cdot \color{blue}{{\left(6.28318530718 \cdot u2\right)}^{0.5}}\right) \]

   4. pow-prod-down 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)\right)}^{0.5}\right)} \]

   5. swap-sqr 98.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\color{blue}{\left(\left(6.28318530718 \cdot 6.28318530718\right) \cdot \left(u2 \cdot u2\right)\right)}}^{0.5}\right) \]

   6. metadata-eval 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

   2. associate-*r* 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{\left(39.47841760436263 \cdot u2\right) \cdot u2}}\right) \]

   3. *-commutative 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{\color{blue}{u2 \cdot \left(39.47841760436263 \cdot u2\right)}}\right) \]

   4. *-commutative 98.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(\sqrt{u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}}\right) \]

5. Simplified 98.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \cos \color{blue}{\left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right)} \]
6. Step-by-step derivation

   1. clear-num 98.9%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   2. inv-pow 98.9%

      \[\leadsto \sqrt{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{-1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   3. div-sub 98.8%

      \[\leadsto \sqrt{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   4. pow1 98.8%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} - \frac{\color{blue}{{u1}^{1}}}{u1}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   5. pow1 98.8%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} - \frac{{u1}^{1}}{\color{blue}{{u1}^{1}}}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   6. pow-div 98.8%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} - \color{blue}{{u1}^{\left(1 - 1\right)}}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   7. metadata-eval 98.8%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   8. metadata-eval 98.8%

      \[\leadsto \sqrt{{\left(\frac{1}{u1} - \color{blue}{1}\right)}^{-1}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{{\left(\frac{1}{u1} - 1\right)}^{-1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]
8. Step-by-step derivation

   1. unpow-1 98.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} - 1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   2. sub-neg 98.8%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

   3. metadata-eval 98.8%

      \[\leadsto \sqrt{\frac{1}{\frac{1}{u1} + \color{blue}{-1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

9. Simplified 98.8%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} + -1}}} \cdot \cos \left(\sqrt{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}\right) \]

10. Taylor expanded in u2 around 0 77.1%

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

    1. sub-neg 77.1%

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

    2. metadata-eval 77.1%

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

    3. unpow-1 77.1%

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

    4. metadata-eval 77.1%

       \[\leadsto \sqrt{{\left(\frac{1}{u1} + -1\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

    5. pow-sqr 77.2%

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

    6. rem-sqrt-square 77.2%

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

    7. sqr-pow 76.7%

       \[\leadsto \left|\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}}\right| \]

    8. fabs-sqr 76.7%

       \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-0.5}{2}\right)}} \]

    9. sqr-pow 77.2%

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

    10. +-commutative 77.2%

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

12. Simplified 77.2%

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

13. Final simplification 77.2%

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

Alternative 8: 79.9% 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 98.8%

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

2. Taylor expanded in u2 around 0 77.1%

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

3. Final simplification 77.1%

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

Alternative 9: 63.5% 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 98.8%

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

2. Taylor expanded in u2 around 0 77.1%

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

3. Taylor expanded in u1 around 0 61.1%

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

4. Final simplification 61.1%

   \[\leadsto \sqrt{u1} \]

Alternative 10: 19.1% accurate, 209.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = u1
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

2. Taylor expanded in u1 around 0 86.2%

   \[\leadsto \sqrt{\color{blue}{{u1}^{2} + u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
3. Step-by-step derivation

   1. unpow2 86.2%

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

   2. fma-udef 86.2%

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

4. Simplified 86.2%

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

5. Taylor expanded in u1 around inf 20.0%

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

   1. *-commutative 20.0%

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

7. Simplified 20.0%

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

8. Taylor expanded in u2 around 0 19.2%

   \[\leadsto \color{blue}{1} \cdot u1 \]

9. Final simplification 19.2%

   \[\leadsto u1 \]

Reproduce

herbie shell --seed 2023274 
(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))))