Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 98.9%

Time: 16.5s

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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 - u1}{u1}\right)}^{-0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (/ (- 1.0 u1) u1) -0.5) (cos (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f - u1) / u1), -0.5f) * 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 = (((1.0e0 - u1) / u1) ** (-0.5e0)) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(Float32(Float32(1.0) - u1) / u1) ^ Float32(-0.5)) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) - u1) / u1) ^ single(-0.5)) * cos((single(6.28318530718) * u2));
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. clear-num 98.8%

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

   2. sqrt-div 98.5%

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

   3. metadata-eval 98.5%

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

   4. div-sub 98.4%

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

   5. inv-pow 98.4%

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

   6. pow1 98.4%

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

   7. pow1 98.4%

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

   8. pow-div 98.4%

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

   9. metadata-eval 98.4%

      \[\leadsto \frac{1}{\sqrt{{u1}^{-1} - {u1}^{\color{blue}{0}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   10. metadata-eval 98.4%

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

3. Applied egg-rr 98.4%

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

   1. unpow-1 98.4%

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

   2. sub-neg 98.4%

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

   3. metadata-eval 98.4%

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

5. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{1}{u1} + -1}}\right)\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. expm1-udef 80.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{1}{u1} + -1}}\right)} - 1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   3. inv-pow 80.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{1}{u1} + -1}\right)}^{-1}}\right)} - 1\right) \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   4. sqrt-pow2 80.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{u1} + -1\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   5. metadata-eval 80.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \cos \left(6.28318530718 \cdot u2\right) \]

7. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)} - 1\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
8. Step-by-step derivation

   1. expm1-def 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   2. expm1-log1p 98.8%

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

   3. metadata-eval 98.8%

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

   4. sub-neg 98.8%

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

   5. *-inverses 98.8%

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

   6. div-sub 98.9%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

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

Alternative 2: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(6.28318530718 \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9932000041007996:\\ \;\;\;\;t_0 \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* 6.28318530718 u2))))
   (if (<= t_0 0.9932000041007996)
     (* t_0 (sqrt (* u1 (+ 1.0 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((6.28318530718f * u2));
	float tmp;
	if (t_0 <= 0.9932000041007996f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + 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((6.28318530718e0 * u2))
    if (t_0 <= 0.9932000041007996e0) then
        tmp = t_0 * sqrt((u1 * (1.0e0 + 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(Float32(6.28318530718) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9932000041007996))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	else
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	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(0.9932000041007996))
		tmp = t_0 * sqrt((u1 * (single(1.0) + 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)) < 0.993200004

   1. Initial program 97.1%

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

      1. clear-num 97.2%

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

      2. associate-/r/ 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in u1 around 0 87.0%

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

   if 0.993200004 < (cos.f32 (*.f32 314159265359/50000000000 u2))

   1. Initial program 99.1%

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

   2. Taylor expanded in u2 around 0 98.6%

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

      1. *-rgt-identity 98.6%

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

      2. *-commutative 98.6%

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

      3. associate-*l* 98.6%

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

      4. distribute-lft-out 98.5%

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

      5. unpow2 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(6.28318530718 \cdot u2\right) \leq 0.9932000041007996:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \end{array} \]

Alternative 3: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3050000071525574:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.3050000071525574)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (/ (cos (* 6.28318530718 u2)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.3050000071525574f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} else {
		tmp = cosf((6.28318530718f * u2)) / sqrtf((1.0f / 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) <= 0.3050000071525574e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = cos((6.28318530718e0 * u2)) / sqrt((1.0e0 / 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(0.3050000071525574))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	else
		tmp = Float32(cos(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.3050000071525574))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	else
		tmp = cos((single(6.28318530718) * u2)) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.305000007

   1. Initial program 99.1%

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

   2. Taylor expanded in u2 around 0 97.5%

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

      1. *-rgt-identity 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*l* 97.5%

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

      4. distribute-lft-out 97.4%

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

      5. unpow2 97.4%

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

   4. Simplified 97.4%

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

   if 0.305000007 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 96.6%

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

      1. clear-num 96.6%

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

      2. sqrt-div 96.7%

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

      3. metadata-eval 96.7%

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

      4. div-sub 96.7%

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

      5. inv-pow 96.7%

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

      6. pow1 96.7%

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

      7. pow1 96.7%

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

      8. pow-div 96.7%

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

      9. metadata-eval 96.7%

         \[\leadsto \frac{1}{\sqrt{{u1}^{-1} - {u1}^{\color{blue}{0}}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      10. metadata-eval 96.7%

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

   3. Applied egg-rr 96.7%

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

      1. unpow-1 96.7%

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

      2. sub-neg 96.7%

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

      3. metadata-eval 96.7%

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

   5. Simplified 96.7%

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

      1. associate-*l/ 96.9%

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

      2. *-un-lft-identity 96.9%

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

      3. add-sqr-sqrt 73.9%

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

      4. add-sqr-sqrt 96.9%

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

      5. *-commutative 96.9%

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

   7. Applied egg-rr 96.9%

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

   8. Taylor expanded in u1 around 0 74.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3050000071525574:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(6.28318530718 \cdot u2\right) \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (cos (* 6.28318530718 u2)) (pow (+ (/ 1.0 u1) -1.0) -0.5)))

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((6.28318530718f * u2)) * 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 = cos((6.28318530718e0 * u2)) * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cos(Float32(Float32(6.28318530718) * u2)) * (Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = cos((single(6.28318530718) * u2)) * (((single(1.0) / u1) + single(-1.0)) ^ 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. pow1/2 98.8%

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

   2. clear-num 98.8%

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

   3. inv-pow 98.8%

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

   4. pow-pow 98.9%

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

   5. div-sub 98.8%

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

   6. inv-pow 98.8%

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

   7. pow1 98.8%

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

   8. pow1 98.8%

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

   9. pow-div 98.8%

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

   10. metadata-eval 98.8%

       \[\leadsto {\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 0.5\right)} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   11. metadata-eval 98.8%

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

   12. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{{\left({u1}^{-1} - 1\right)}^{-0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
4. Step-by-step derivation

   1. unpow-1 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 5: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3050000071525574:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.3050000071525574)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.3050000071525574f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} 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) <= 0.3050000071525574e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    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(0.3050000071525574))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	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(0.3050000071525574))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	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) < 0.305000007

   1. Initial program 99.1%

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

   2. Taylor expanded in u2 around 0 97.5%

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

      1. *-rgt-identity 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*l* 97.5%

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

      4. distribute-lft-out 97.4%

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

      5. unpow2 97.4%

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

   4. Simplified 97.4%

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

   if 0.305000007 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 96.6%

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

   2. Taylor expanded in u1 around 0 74.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3050000071525574:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return cosf((6.28318530718f * 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 = cos((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = cos((single(6.28318530718) * u2)) * 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. Final simplification 98.8%

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

Alternative 7: 84.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 / (1.0f - u1)) * (1.0f + ((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)) * (1.0e0 + ((u2 * u2) * (-39.47841760436263e0)))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 / (single(1.0) - u1)) * (single(1.0) + ((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 95.1%

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

   2. pow1/2 95.1%

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

   3. pow1/2 95.1%

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

   4. pow-prod-down 96.2%

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

   5. swap-sqr 96.2%

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

   6. add-sqr-sqrt 96.3%

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

   7. pow2 96.3%

      \[\leadsto {\left(\frac{u1}{1 - u1} \cdot \color{blue}{{\cos \left(6.28318530718 \cdot u2\right)}^{2}}\right)}^{0.5} \]

3. Applied egg-rr 96.3%

   \[\leadsto \color{blue}{{\left(\frac{u1}{1 - u1} \cdot {\cos \left(6.28318530718 \cdot u2\right)}^{2}\right)}^{0.5}} \]
4. Step-by-step derivation

   1. unpow1/2 96.3%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot {\cos \left(6.28318530718 \cdot u2\right)}^{2}}} \]

5. Simplified 96.3%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot {\cos \left(6.28318530718 \cdot u2\right)}^{2}}} \]

6. Taylor expanded in u2 around 0 87.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(1 + -39.47841760436263 \cdot {u2}^{2}\right)}} \]
7. Step-by-step derivation

   1. *-commutative 87.8%

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

   2. unpow2 87.8%

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

8. Simplified 87.8%

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

9. Final simplification 87.8%

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

Alternative 8: 87.7% accurate, 1.8× speedup

Math

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

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(Float32(u2 * 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. Taylor expanded in u2 around 0 90.0%

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

   1. *-rgt-identity 90.0%

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

   2. *-commutative 90.0%

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

   3. associate-*l* 90.0%

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

   4. distribute-lft-out 89.9%

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

   5. unpow2 89.9%

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

4. Simplified 89.9%

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

5. Final simplification 89.9%

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

Alternative 9: 79.5% 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 81.8%

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

3. Final simplification 81.8%

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

Alternative 10: 62.8% 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 81.8%

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

3. Taylor expanded in u1 around 0 65.1%

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

4. Final simplification 65.1%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023285 
(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))))