Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.5s

Alternatives: 13

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

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((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))) * sin((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((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))) * sin((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.5%

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

   1. pow1/2 98.5%

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

   2. sqr-pow 98.1%

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

   3. clear-num 98.2%

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

   4. inv-pow 98.2%

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

   5. metadata-eval 98.2%

      \[\leadsto \left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{0.25}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. metadata-eval 98.2%

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

   7. pow-pow 98.1%

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

   8. metadata-eval 98.1%

      \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \color{blue}{0.25}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. metadata-eval 98.1%

      \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{\color{blue}{-0.25}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   10. clear-num 98.1%

       \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   11. inv-pow 98.1%

       \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   12. metadata-eval 98.1%

       \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{0.25}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   13. metadata-eval 98.1%

       \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{\left(0.5 \cdot 0.5\right)}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   14. pow-pow 98.0%

       \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \left(0.5 \cdot 0.5\right)\right)}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   15. metadata-eval 98.0%

       \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \color{blue}{0.25}\right)}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   16. metadata-eval 98.0%

       \[\leadsto \left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left(\frac{1 - u1}{u1}\right)}^{\color{blue}{-0.25}}\right) \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left(\frac{1 - u1}{u1}\right)}^{-0.25}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
4. Step-by-step derivation

   1. pow-sqr 98.5%

      \[\leadsto \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(2 \cdot -0.25\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. metadata-eval 98.5%

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

   3. div-sub 98.5%

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

   4. sub-neg 98.5%

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

   5. *-inverses 98.5%

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

   6. metadata-eval 98.5%

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

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

Alternative 2: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.20000000298023224:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + u2 \cdot \left(u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.20000000298023224)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (+ (* 6.28318530718 u2) (* u2 (* u2 (* u2 -41.341702240407926)))))
   (* (sin (* 6.28318530718 u2)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.20000000298023224f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * ((6.28318530718f * u2) + (u2 * (u2 * (u2 * -41.341702240407926f))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 + (u1 * 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) <= 0.20000000298023224e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * ((6.28318530718e0 * u2) + (u2 * (u2 * (u2 * (-41.341702240407926e0)))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 + (u1 * 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(0.20000000298023224))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(Float32(6.28318530718) * u2) + Float32(u2 * Float32(u2 * Float32(u2 * Float32(-41.341702240407926))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.20000000298023224))
		tmp = sqrt((u1 / (single(1.0) - u1))) * ((single(6.28318530718) * u2) + (u2 * (u2 * (u2 * single(-41.341702240407926)))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 + (u1 * u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.200000003

   1. Initial program 98.5%

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

   2. Taylor expanded in u2 around 0 98.0%

      \[\leadsto \color{blue}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 98.0%

         \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      2. associate-*r* 98.0%

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-41.341702240407926 \cdot {u2}^{3}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. distribute-rgt-out 98.0%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot {u2}^{3}\right)} \]

      4. unpow3 98.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

      5. associate-*r* 98.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2}\right) \]

      6. distribute-rgt-out 98.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

   4. Simplified 98.0%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 98.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(6.28318530718 \cdot u2 + \left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right)} \]

      2. associate-*r* 98.0%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(\left(-41.341702240407926 \cdot u2\right) \cdot u2\right)} \cdot u2\right) \]

   6. Applied egg-rr 98.0%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(6.28318530718 \cdot u2 + \left(\left(-41.341702240407926 \cdot u2\right) \cdot u2\right) \cdot u2\right)} \]

   if 0.200000003 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.4%

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

      1. clear-num 98.1%

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

      2. associate-/r/ 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in u1 around 0 89.8%

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

      1. +-commutative 89.8%

         \[\leadsto \sqrt{\color{blue}{u1 + {u1}^{2}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

      2. unpow2 89.8%

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

   6. Simplified 89.8%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.20000000298023224:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + u2 \cdot \left(u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.23999999463558197:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + u2 \cdot \left(u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.23999999463558197)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (+ (* 6.28318530718 u2) (* u2 (* u2 (* u2 -41.341702240407926)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.23999999463558197f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * ((6.28318530718f * u2) + (u2 * (u2 * (u2 * -41.341702240407926f))));
	} else {
		tmp = sinf((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) <= 0.23999999463558197e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * ((6.28318530718e0 * u2) + (u2 * (u2 * (u2 * (-41.341702240407926e0)))))
    else
        tmp = sin((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(0.23999999463558197))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(Float32(6.28318530718) * u2) + Float32(u2 * Float32(u2 * Float32(u2 * Float32(-41.341702240407926))))));
	else
		tmp = Float32(sin(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(0.23999999463558197))
		tmp = sqrt((u1 / (single(1.0) - u1))) * ((single(6.28318530718) * u2) + (u2 * (u2 * (u2 * single(-41.341702240407926)))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.239999995

   1. Initial program 98.5%

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

   2. Taylor expanded in u2 around 0 97.9%

      \[\leadsto \color{blue}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 97.9%

         \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      2. associate-*r* 97.9%

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-41.341702240407926 \cdot {u2}^{3}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. distribute-rgt-out 97.9%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot {u2}^{3}\right)} \]

      4. unpow3 97.9%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

      5. associate-*r* 97.9%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2}\right) \]

      6. distribute-rgt-out 97.9%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

   4. Simplified 97.9%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 97.9%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(6.28318530718 \cdot u2 + \left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right)} \]

      2. associate-*r* 97.9%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(\left(-41.341702240407926 \cdot u2\right) \cdot u2\right)} \cdot u2\right) \]

   6. Applied egg-rr 97.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(6.28318530718 \cdot u2 + \left(\left(-41.341702240407926 \cdot u2\right) \cdot u2\right) \cdot u2\right)} \]

   if 0.239999995 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.4%

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

   2. Taylor expanded in u1 around 0 79.3%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.23999999463558197:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + u2 \cdot \left(u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Final simplification 98.5%

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

Alternative 5: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + u2 \cdot \left(u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (+ (* 6.28318530718 u2) (* u2 (* u2 (* u2 -41.341702240407926))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * ((6.28318530718f * u2) + (u2 * (u2 * (u2 * -41.341702240407926f))));
}

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))) * ((6.28318530718e0 * u2) + (u2 * (u2 * (u2 * (-41.341702240407926e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(Float32(6.28318530718) * u2) + Float32(u2 * Float32(u2 * Float32(u2 * Float32(-41.341702240407926))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * ((single(6.28318530718) * u2) + (u2 * (u2 * (u2 * single(-41.341702240407926)))));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 89.2%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
3. Step-by-step derivation

   1. associate-*r* 89.3%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   2. associate-*r* 89.3%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-41.341702240407926 \cdot {u2}^{3}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. distribute-rgt-out 89.2%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot {u2}^{3}\right)} \]

   4. unpow3 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

   5. associate-*r* 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2}\right) \]

   6. distribute-rgt-out 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

4. Simplified 89.2%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]
5. Step-by-step derivation

   1. distribute-rgt-in 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(6.28318530718 \cdot u2 + \left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2\right)} \]

   2. associate-*r* 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(\left(-41.341702240407926 \cdot u2\right) \cdot u2\right)} \cdot u2\right) \]

6. Applied egg-rr 89.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(6.28318530718 \cdot u2 + \left(\left(-41.341702240407926 \cdot u2\right) \cdot u2\right) \cdot u2\right)} \]

7. Final simplification 89.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + u2 \cdot \left(u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right) \]

Alternative 6: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.023000000044703484:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.023000000044703484)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt u1) (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.023000000044703484f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * u2))));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.023000000044703484e0) then
        tmp = (6.28318530718e0 * u2) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(u1) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.023000000044703484))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.023000000044703484))
		tmp = (single(6.28318530718) * u2) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(u1) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.023

   1. Initial program 98.5%

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

   2. Taylor expanded in u2 around 0 95.0%

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

      1. associate-*r* 95.1%

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

   4. Simplified 95.1%

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

   if 0.023 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 59.4%

      \[\leadsto \color{blue}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* 59.3%

         \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

      2. associate-*r* 59.3%

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-41.341702240407926 \cdot {u2}^{3}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. distribute-rgt-out 59.4%

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot {u2}^{3}\right)} \]

      4. unpow3 59.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

      5. associate-*r* 59.4%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2}\right) \]

      6. distribute-rgt-out 59.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

   4. Simplified 59.2%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

   5. Taylor expanded in u1 around 0 52.4%

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.023000000044703484:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)\\ \end{array} \]

Alternative 7: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (* u2 (+ 6.28318530718 (* -41.341702240407926 (* u2 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (-41.341702240407926f * (u2 * 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))) * (u2 * (6.28318530718e0 + ((-41.341702240407926e0) * (u2 * u2))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(-41.341702240407926) * Float32(u2 * u2)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (single(-41.341702240407926) * (u2 * u2))));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 89.2%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
3. Step-by-step derivation

   1. associate-*r* 89.3%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   2. associate-*r* 89.3%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-41.341702240407926 \cdot {u2}^{3}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. distribute-rgt-out 89.2%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot {u2}^{3}\right)} \]

   4. unpow3 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

   5. associate-*r* 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2}\right) \]

   6. distribute-rgt-out 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

4. Simplified 89.2%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

5. Final simplification 89.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \]

Alternative 8: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (* u2 (+ 6.28318530718 (* u2 (* u2 -41.341702240407926))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + (u2 * (u2 * -41.341702240407926f))));
}

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))) * (u2 * (6.28318530718e0 + (u2 * (u2 * (-41.341702240407926e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(u2 * Float32(u2 * Float32(-41.341702240407926))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + (u2 * (u2 * single(-41.341702240407926)))));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 89.2%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
3. Step-by-step derivation

   1. associate-*r* 89.3%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} + -41.341702240407926 \cdot \left({u2}^{3} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]

   2. associate-*r* 89.3%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\left(-41.341702240407926 \cdot {u2}^{3}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. distribute-rgt-out 89.2%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot {u2}^{3}\right)} \]

   4. unpow3 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + -41.341702240407926 \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

   5. associate-*r* 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2 + \color{blue}{\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right)\right) \cdot u2}\right) \]

   6. distribute-rgt-out 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

4. Simplified 89.2%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

5. Taylor expanded in u2 around 0 89.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \color{blue}{-41.341702240407926 \cdot {u2}^{2}}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \color{blue}{{u2}^{2} \cdot -41.341702240407926}\right)\right) \]

   2. unpow2 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \color{blue}{\left(u2 \cdot u2\right)} \cdot -41.341702240407926\right)\right) \]

   3. associate-*r* 89.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \color{blue}{u2 \cdot \left(u2 \cdot -41.341702240407926\right)}\right)\right) \]

7. Simplified 89.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \color{blue}{u2 \cdot \left(u2 \cdot -41.341702240407926\right)}\right)\right) \]

8. Final simplification 89.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right) \]

Alternative 9: 72.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 + u1 \cdot u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * sqrtf((u1 + (u1 * u1))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * sqrt((u1 + (u1 * u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 + Float32(u1 * u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt((u1 + (u1 * u1))));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 81.5%

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

   1. flip-- 81.5%

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

   2. associate-/r/ 81.5%

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

   3. metadata-eval 81.5%

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

   4. +-commutative 81.5%

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

4. Applied egg-rr 81.5%

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

5. Taylor expanded in u1 around 0 72.8%

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

   1. +-commutative 72.8%

      \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\color{blue}{u1 + {u1}^{2}}}\right) \]

   2. unpow2 72.8%

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

7. Simplified 72.8%

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

8. Final simplification 72.8%

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

Alternative 10: 81.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 81.5%

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

3. Final simplification 81.5%

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

Alternative 11: 81.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 81.5%

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

   1. associate-*r* 81.6%

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

4. Simplified 81.6%

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

5. Final simplification 81.6%

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

Alternative 12: 64.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * 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 = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(u1)))
end

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 81.5%

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

3. Taylor expanded in u1 around 0 64.1%

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

4. Final simplification 64.1%

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

Alternative 13: 64.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (* 6.28318530718 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (6.28318530718f * 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 = u2 * (6.28318530718e0 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(6.28318530718) * sqrt(u1)))
end

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 81.5%

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

3. Taylor expanded in u1 around 0 64.1%

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

   1. expm1-log1p-u 64.1%

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

   2. expm1-udef 25.8%

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

5. Applied egg-rr 25.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 64.1%

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

   2. expm1-log1p 64.1%

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

   3. *-commutative 64.1%

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

   4. associate-*l* 64.1%

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

7. Simplified 64.1%

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

8. Final simplification 64.1%

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

Reproduce

herbie shell --seed 2023252 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :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))) (sin (* 6.28318530718 u2))))