quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.5% → 86.0%

Time: 14.1s

Alternatives: 9

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 51.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{c \cdot \left(0.5 + 0.125 \cdot \left(c \cdot \frac{a}{{b\_2}^{2}}\right)\right)}{-b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{{b\_2}^{2}}{-b\_2} - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.6e-68)
   (/ (* c (+ 0.5 (* 0.125 (* c (/ a (pow b_2 2.0)))))) (- b_2))
   (if (<= b_2 6e+110)
     (/ (- (/ (pow b_2 2.0) (- b_2)) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.6e-68) {
		tmp = (c * (0.5 + (0.125 * (c * (a / pow(b_2, 2.0)))))) / -b_2;
	} else if (b_2 <= 6e+110) {
		tmp = ((pow(b_2, 2.0) / -b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7.6d-68)) then
        tmp = (c * (0.5d0 + (0.125d0 * (c * (a / (b_2 ** 2.0d0)))))) / -b_2
    else if (b_2 <= 6d+110) then
        tmp = (((b_2 ** 2.0d0) / -b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.6e-68) {
		tmp = (c * (0.5 + (0.125 * (c * (a / Math.pow(b_2, 2.0)))))) / -b_2;
	} else if (b_2 <= 6e+110) {
		tmp = ((Math.pow(b_2, 2.0) / -b_2) - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.6e-68:
		tmp = (c * (0.5 + (0.125 * (c * (a / math.pow(b_2, 2.0)))))) / -b_2
	elif b_2 <= 6e+110:
		tmp = ((math.pow(b_2, 2.0) / -b_2) - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.6e-68)
		tmp = Float64(Float64(c * Float64(0.5 + Float64(0.125 * Float64(c * Float64(a / (b_2 ^ 2.0)))))) / Float64(-b_2));
	elseif (b_2 <= 6e+110)
		tmp = Float64(Float64(Float64((b_2 ^ 2.0) / Float64(-b_2)) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.6e-68)
		tmp = (c * (0.5 + (0.125 * (c * (a / (b_2 ^ 2.0)))))) / -b_2;
	elseif (b_2 <= 6e+110)
		tmp = (((b_2 ^ 2.0) / -b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.6e-68], N[(N[(c * N[(0.5 + N[(0.125 * N[(c * N[(a / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b$95$2)), $MachinePrecision], If[LessEqual[b$95$2, 6e+110], N[(N[(N[(N[Power[b$95$2, 2.0], $MachinePrecision] / (-b$95$2)), $MachinePrecision] - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.60000000000000075e-68

   1. Initial program 15.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fmm-def 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fmm-def 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. sub-neg 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 + \left(-a \cdot c\right)\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. +-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\left(-a \cdot c\right) + b\_2 \cdot b\_2\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. distribute-rgt-neg-in 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{a \cdot \left(-c\right)} + b\_2 \cdot b\_2\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. fma-define 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. pow2 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, \color{blue}{{b\_2}^{2}}\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. *-commutative 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. fma-undefine 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      15. distribute-lft-neg-in 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      16. *-commutative 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      17. distribute-rgt-neg-in 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      18. fma-define 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      19. *-commutative 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

   4. Applied egg-rr 15.0%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   5. Step-by-step derivation

      1. fma-define 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(a \cdot \left(-c\right) + {b\_2}^{2}\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      2. +-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} + a \cdot \left(-c\right)\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      3. *-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} + \color{blue}{\left(-c\right) \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      4. cancel-sign-sub-inv 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - c \cdot a\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      5. count-2 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

      6. *-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   6. Simplified 15.0%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

   7. Taylor expanded in b_2 around -inf 69.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{-0.5 \cdot \frac{2 \cdot \left(-1 \cdot \left(a \cdot c\right) + a \cdot c\right) - a \cdot c}{a} + 0.125 \cdot \frac{{\left(2 \cdot \left(-1 \cdot \left(a \cdot c\right) + a \cdot c\right) - a \cdot c\right)}^{2}}{a \cdot {b\_2}^{2}}}{b\_2}} \]

   8. Taylor expanded in c around 0 68.9%

      \[\leadsto -1 \cdot \frac{\color{blue}{c \cdot \left(-0.5 \cdot \frac{2 \cdot \left(a + -1 \cdot a\right) - a}{a} + 0.125 \cdot \frac{c \cdot {\left(2 \cdot \left(a + -1 \cdot a\right) - a\right)}^{2}}{a \cdot {b\_2}^{2}}\right)}}{b\_2} \]

   10. Simplified 85.4%

       \[\leadsto -1 \cdot \frac{\color{blue}{c \cdot \left(0.5 + 0.125 \cdot \left(c \cdot \left(1 \cdot \frac{a}{{b\_2}^{2}}\right)\right)\right)}}{b\_2} \]

   if -7.60000000000000075e-68 < b_2 < 6.00000000000000014e110

   1. Initial program 79.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. neg-sub0 79.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\_2\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. flip-- 79.9%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b\_2 \cdot b\_2}{0 + b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. metadata-eval 79.9%

         \[\leadsto \frac{\frac{\color{blue}{0} - b\_2 \cdot b\_2}{0 + b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. pow2 79.9%

         \[\leadsto \frac{\frac{0 - \color{blue}{{b\_2}^{2}}}{0 + b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. add-sqr-sqrt 54.1%

         \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{0 + \color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. sqrt-prod 52.7%

         \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{0 + \color{blue}{\sqrt{b\_2 \cdot b\_2}}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. sqr-neg 52.7%

         \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{0 + \sqrt{\color{blue}{\left(-b\_2\right) \cdot \left(-b\_2\right)}}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      8. sqrt-unprod 25.7%

         \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{0 + \color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      9. add-sqr-sqrt 49.5%

         \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{0 + \color{blue}{\left(-b\_2\right)}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      10. sub-neg 49.5%

          \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{\color{blue}{0 - b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      11. neg-sub0 49.5%

          \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{\color{blue}{-b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      12. add-sqr-sqrt 25.7%

          \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      13. sqrt-unprod 52.7%

          \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      14. sqr-neg 52.7%

          \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{\sqrt{\color{blue}{b\_2 \cdot b\_2}}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      15. sqrt-prod 54.1%

          \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      16. add-sqr-sqrt 79.9%

          \[\leadsto \frac{\frac{0 - {b\_2}^{2}}{\color{blue}{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 79.9%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b\_2}^{2}}{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   5. Step-by-step derivation

      1. sub0-neg 79.9%

         \[\leadsto \frac{\frac{\color{blue}{-{b\_2}^{2}}}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   6. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{\frac{-{b\_2}^{2}}{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   if 6.00000000000000014e110 < b_2

   1. Initial program 51.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 98.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{c \cdot \left(0.5 + 0.125 \cdot \left(c \cdot \frac{a}{{b\_2}^{2}}\right)\right)}{-b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{{b\_2}^{2}}{-b\_2} - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{c \cdot \left(0.5 + 0.125 \cdot \left(c \cdot \frac{a}{{b\_2}^{2}}\right)\right)}{-b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.25 \cdot 10^{+108}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-69)
   (/ (* c (+ 0.5 (* 0.125 (* c (/ a (pow b_2 2.0)))))) (- b_2))
   (if (<= b_2 2.25e+108)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-69) {
		tmp = (c * (0.5 + (0.125 * (c * (a / pow(b_2, 2.0)))))) / -b_2;
	} else if (b_2 <= 2.25e+108) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.2d-69)) then
        tmp = (c * (0.5d0 + (0.125d0 * (c * (a / (b_2 ** 2.0d0)))))) / -b_2
    else if (b_2 <= 2.25d+108) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-69) {
		tmp = (c * (0.5 + (0.125 * (c * (a / Math.pow(b_2, 2.0)))))) / -b_2;
	} else if (b_2 <= 2.25e+108) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-69:
		tmp = (c * (0.5 + (0.125 * (c * (a / math.pow(b_2, 2.0)))))) / -b_2
	elif b_2 <= 2.25e+108:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-69)
		tmp = Float64(Float64(c * Float64(0.5 + Float64(0.125 * Float64(c * Float64(a / (b_2 ^ 2.0)))))) / Float64(-b_2));
	elseif (b_2 <= 2.25e+108)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.2e-69)
		tmp = (c * (0.5 + (0.125 * (c * (a / (b_2 ^ 2.0)))))) / -b_2;
	elseif (b_2 <= 2.25e+108)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-69], N[(N[(c * N[(0.5 + N[(0.125 * N[(c * N[(a / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b$95$2)), $MachinePrecision], If[LessEqual[b$95$2, 2.25e+108], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.19999999999999999e-69

   1. Initial program 15.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fmm-def 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fmm-def 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. sub-neg 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 + \left(-a \cdot c\right)\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. +-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\left(-a \cdot c\right) + b\_2 \cdot b\_2\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. distribute-rgt-neg-in 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{a \cdot \left(-c\right)} + b\_2 \cdot b\_2\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. fma-define 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. pow2 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, \color{blue}{{b\_2}^{2}}\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. *-commutative 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. fma-undefine 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      15. distribute-lft-neg-in 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      16. *-commutative 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      17. distribute-rgt-neg-in 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      18. fma-define 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      19. *-commutative 15.0%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

   4. Applied egg-rr 15.0%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   5. Step-by-step derivation

      1. fma-define 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(a \cdot \left(-c\right) + {b\_2}^{2}\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      2. +-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} + a \cdot \left(-c\right)\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      3. *-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} + \color{blue}{\left(-c\right) \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      4. cancel-sign-sub-inv 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - c \cdot a\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      5. count-2 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

      6. *-commutative 15.0%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   6. Simplified 15.0%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

   7. Taylor expanded in b_2 around -inf 69.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{-0.5 \cdot \frac{2 \cdot \left(-1 \cdot \left(a \cdot c\right) + a \cdot c\right) - a \cdot c}{a} + 0.125 \cdot \frac{{\left(2 \cdot \left(-1 \cdot \left(a \cdot c\right) + a \cdot c\right) - a \cdot c\right)}^{2}}{a \cdot {b\_2}^{2}}}{b\_2}} \]

   8. Taylor expanded in c around 0 68.9%

      \[\leadsto -1 \cdot \frac{\color{blue}{c \cdot \left(-0.5 \cdot \frac{2 \cdot \left(a + -1 \cdot a\right) - a}{a} + 0.125 \cdot \frac{c \cdot {\left(2 \cdot \left(a + -1 \cdot a\right) - a\right)}^{2}}{a \cdot {b\_2}^{2}}\right)}}{b\_2} \]

   10. Simplified 85.4%

       \[\leadsto -1 \cdot \frac{\color{blue}{c \cdot \left(0.5 + 0.125 \cdot \left(c \cdot \left(1 \cdot \frac{a}{{b\_2}^{2}}\right)\right)\right)}}{b\_2} \]

   if -3.19999999999999999e-69 < b_2 < 2.25e108

   1. Initial program 79.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 2.25e108 < b_2

   1. Initial program 51.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 98.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{c \cdot \left(0.5 + 0.125 \cdot \left(c \cdot \frac{a}{{b\_2}^{2}}\right)\right)}{-b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.25 \cdot 10^{+108}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;c \cdot \left(0.125 \cdot \frac{c \cdot a}{{b\_2}^{3}} + 0.5 \cdot \frac{1}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.35e+154)
   (* c (+ (* 0.125 (/ (* c a) (pow b_2 3.0))) (* 0.5 (/ 1.0 b_2))))
   (if (<= b_2 1.3e+112)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e+154) {
		tmp = c * ((0.125 * ((c * a) / pow(b_2, 3.0))) + (0.5 * (1.0 / b_2)));
	} else if (b_2 <= 1.3e+112) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.35d+154)) then
        tmp = c * ((0.125d0 * ((c * a) / (b_2 ** 3.0d0))) + (0.5d0 * (1.0d0 / b_2)))
    else if (b_2 <= 1.3d+112) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e+154) {
		tmp = c * ((0.125 * ((c * a) / Math.pow(b_2, 3.0))) + (0.5 * (1.0 / b_2)));
	} else if (b_2 <= 1.3e+112) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.35e+154:
		tmp = c * ((0.125 * ((c * a) / math.pow(b_2, 3.0))) + (0.5 * (1.0 / b_2)))
	elif b_2 <= 1.3e+112:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.35e+154)
		tmp = Float64(c * Float64(Float64(0.125 * Float64(Float64(c * a) / (b_2 ^ 3.0))) + Float64(0.5 * Float64(1.0 / b_2))));
	elseif (b_2 <= 1.3e+112)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.35e+154)
		tmp = c * ((0.125 * ((c * a) / (b_2 ^ 3.0))) + (0.5 * (1.0 / b_2)));
	elseif (b_2 <= 1.3e+112)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.35e+154], N[(c * N[(N[(0.125 * N[(N[(c * a), $MachinePrecision] / N[Power[b$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.3e+112], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.35000000000000003e154

   1. Initial program 1.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 1.7%

         \[\leadsto \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. sqrt-unprod 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. sqr-neg 0.0%

         \[\leadsto \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. sqrt-prod 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. add-sqr-sqrt 1.7%

         \[\leadsto \frac{\color{blue}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. add-cube-cbrt 1.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right) \cdot \sqrt[3]{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. pow3 1.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 1.7%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   5. Taylor expanded in c around 0 44.1%

      \[\leadsto \color{blue}{c \cdot \left(0.125 \cdot \frac{a \cdot c}{{b\_2}^{3}} + 0.5 \cdot \frac{1}{b\_2}\right)} \]

   if -1.35000000000000003e154 < b_2 < 1.3e112

   1. Initial program 60.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 1.3e112 < b_2

   1. Initial program 51.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 98.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;c \cdot \left(0.125 \cdot \frac{c \cdot a}{{b\_2}^{3}} + 0.5 \cdot \frac{1}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.35e+154)
   (* c (- (/ (* b_2 2.0) (* c a)) (/ 0.5 b_2)))
   (if (<= b_2 2e+109)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e+154) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else if (b_2 <= 2e+109) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.35d+154)) then
        tmp = c * (((b_2 * 2.0d0) / (c * a)) - (0.5d0 / b_2))
    else if (b_2 <= 2d+109) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.35e+154) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else if (b_2 <= 2e+109) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.35e+154:
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2))
	elif b_2 <= 2e+109:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.35e+154)
		tmp = Float64(c * Float64(Float64(Float64(b_2 * 2.0) / Float64(c * a)) - Float64(0.5 / b_2)));
	elseif (b_2 <= 2e+109)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.35e+154)
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	elseif (b_2 <= 2e+109)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.35e+154], N[(c * N[(N[(N[(b$95$2 * 2.0), $MachinePrecision] / N[(c * a), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2e+109], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.35000000000000003e154

   1. Initial program 1.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 1.7%

         \[\leadsto \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. sqrt-unprod 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. sqr-neg 0.0%

         \[\leadsto \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. sqrt-prod 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. add-sqr-sqrt 1.7%

         \[\leadsto \frac{\color{blue}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. add-cube-cbrt 1.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right) \cdot \sqrt[3]{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. pow3 1.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 1.7%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   5. Taylor expanded in b_2 around -inf 1.8%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 1.8%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}}{a} \]

      2. neg-mul-1 1.8%

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right)} \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}{a} \]

      3. rem-cube-cbrt 1.8%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-1}\right) - 1\right)}{a} \]

      4. associate--l+ 1.8%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \color{blue}{\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(-1 - 1\right)\right)}}{a} \]

      5. metadata-eval 1.8%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-2}\right)}{a} \]

   7. Simplified 1.8%

      \[\leadsto \frac{\color{blue}{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + -2\right)}}{a} \]

   8. Taylor expanded in c around inf 16.0%

      \[\leadsto \color{blue}{c \cdot \left(2 \cdot \frac{b\_2}{a \cdot c} - 0.5 \cdot \frac{1}{b\_2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 16.0%

         \[\leadsto c \cdot \left(\color{blue}{\frac{2 \cdot b\_2}{a \cdot c}} - 0.5 \cdot \frac{1}{b\_2}\right) \]

      2. associate-*r/ 16.0%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \color{blue}{\frac{0.5 \cdot 1}{b\_2}}\right) \]

      3. metadata-eval 16.0%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{\color{blue}{0.5}}{b\_2}\right) \]

   10. Simplified 16.0%

       \[\leadsto \color{blue}{c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{0.5}{b\_2}\right)} \]

   if -1.35000000000000003e154 < b_2 < 1.99999999999999996e109

   1. Initial program 60.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 1.99999999999999996e109 < b_2

   1. Initial program 51.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 98.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{+131}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-81}:\\ \;\;\;\;\frac{b\_2 - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e+131)
   (* c (- (/ (* b_2 2.0) (* c a)) (/ 0.5 b_2)))
   (if (<= b_2 1.35e-81)
     (/ (- b_2 (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e+131) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else if (b_2 <= 1.35e-81) {
		tmp = (b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.6d+131)) then
        tmp = c * (((b_2 * 2.0d0) / (c * a)) - (0.5d0 / b_2))
    else if (b_2 <= 1.35d-81) then
        tmp = (b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e+131) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else if (b_2 <= 1.35e-81) {
		tmp = (b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e+131:
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2))
	elif b_2 <= 1.35e-81:
		tmp = (b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e+131)
		tmp = Float64(c * Float64(Float64(Float64(b_2 * 2.0) / Float64(c * a)) - Float64(0.5 / b_2)));
	elseif (b_2 <= 1.35e-81)
		tmp = Float64(Float64(b_2 - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e+131)
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	elseif (b_2 <= 1.35e-81)
		tmp = (b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e+131], N[(c * N[(N[(N[(b$95$2 * 2.0), $MachinePrecision] / N[(c * a), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.35e-81], N[(N[(b$95$2 - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6e131

   1. Initial program 6.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 4.0%

         \[\leadsto \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. sqrt-unprod 4.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. sqr-neg 4.8%

         \[\leadsto \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. sqrt-prod 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. add-sqr-sqrt 1.9%

         \[\leadsto \frac{\color{blue}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. add-cube-cbrt 1.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right) \cdot \sqrt[3]{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. pow3 1.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 1.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   5. Taylor expanded in b_2 around -inf 1.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 1.9%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}}{a} \]

      2. neg-mul-1 1.9%

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right)} \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}{a} \]

      3. rem-cube-cbrt 1.9%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-1}\right) - 1\right)}{a} \]

      4. associate--l+ 1.9%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \color{blue}{\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(-1 - 1\right)\right)}}{a} \]

      5. metadata-eval 1.9%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-2}\right)}{a} \]

   7. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + -2\right)}}{a} \]

   8. Taylor expanded in c around inf 13.9%

      \[\leadsto \color{blue}{c \cdot \left(2 \cdot \frac{b\_2}{a \cdot c} - 0.5 \cdot \frac{1}{b\_2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 13.9%

         \[\leadsto c \cdot \left(\color{blue}{\frac{2 \cdot b\_2}{a \cdot c}} - 0.5 \cdot \frac{1}{b\_2}\right) \]

      2. associate-*r/ 13.9%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \color{blue}{\frac{0.5 \cdot 1}{b\_2}}\right) \]

      3. metadata-eval 13.9%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{\color{blue}{0.5}}{b\_2}\right) \]

   10. Simplified 13.9%

       \[\leadsto \color{blue}{c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{0.5}{b\_2}\right)} \]

   if -2.6e131 < b_2 < 1.34999999999999995e-81

   1. Initial program 50.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. neg-sub0 50.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\_2\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. sub-neg 50.8%

         \[\leadsto \frac{\color{blue}{\left(0 + \left(-b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. add-sqr-sqrt 32.1%

         \[\leadsto \frac{\left(0 + \color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}}\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. sqrt-unprod 49.2%

         \[\leadsto \frac{\left(0 + \color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}}\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. sqr-neg 49.2%

         \[\leadsto \frac{\left(0 + \sqrt{\color{blue}{b\_2 \cdot b\_2}}\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. sqrt-prod 15.9%

         \[\leadsto \frac{\left(0 + \color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}}\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. add-sqr-sqrt 46.7%

         \[\leadsto \frac{\left(0 + \color{blue}{b\_2}\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 46.7%

      \[\leadsto \frac{\color{blue}{\left(0 + b\_2\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   5. Step-by-step derivation

      1. +-lft-identity 46.7%

         \[\leadsto \frac{\color{blue}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   6. Simplified 46.7%

      \[\leadsto \frac{\color{blue}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   if 1.34999999999999995e-81 < b_2

   1. Initial program 71.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 89.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{+131}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-81}:\\ \;\;\;\;\frac{b\_2 - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{+131}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 2.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b\_2}{-a} - \frac{\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.5e+131)
   (* c (- (/ (* b_2 2.0) (* c a)) (/ 0.5 b_2)))
   (if (<= b_2 2.5e-80)
     (- (/ b_2 (- a)) (/ (sqrt (* c (- a))) a))
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.5e+131) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else if (b_2 <= 2.5e-80) {
		tmp = (b_2 / -a) - (sqrt((c * -a)) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9.5d+131)) then
        tmp = c * (((b_2 * 2.0d0) / (c * a)) - (0.5d0 / b_2))
    else if (b_2 <= 2.5d-80) then
        tmp = (b_2 / -a) - (sqrt((c * -a)) / a)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.5e+131) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else if (b_2 <= 2.5e-80) {
		tmp = (b_2 / -a) - (Math.sqrt((c * -a)) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9.5e+131:
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2))
	elif b_2 <= 2.5e-80:
		tmp = (b_2 / -a) - (math.sqrt((c * -a)) / a)
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9.5e+131)
		tmp = Float64(c * Float64(Float64(Float64(b_2 * 2.0) / Float64(c * a)) - Float64(0.5 / b_2)));
	elseif (b_2 <= 2.5e-80)
		tmp = Float64(Float64(b_2 / Float64(-a)) - Float64(sqrt(Float64(c * Float64(-a))) / a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9.5e+131)
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	elseif (b_2 <= 2.5e-80)
		tmp = (b_2 / -a) - (sqrt((c * -a)) / a);
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9.5e+131], N[(c * N[(N[(N[(b$95$2 * 2.0), $MachinePrecision] / N[(c * a), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.5e-80], N[(N[(b$95$2 / (-a)), $MachinePrecision] - N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.50000000000000015e131

   1. Initial program 6.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 4.0%

         \[\leadsto \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. sqrt-unprod 4.8%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. sqr-neg 4.8%

         \[\leadsto \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. sqrt-prod 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. add-sqr-sqrt 1.9%

         \[\leadsto \frac{\color{blue}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. add-cube-cbrt 1.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right) \cdot \sqrt[3]{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. pow3 1.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 1.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   5. Taylor expanded in b_2 around -inf 1.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 1.9%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}}{a} \]

      2. neg-mul-1 1.9%

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right)} \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}{a} \]

      3. rem-cube-cbrt 1.9%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-1}\right) - 1\right)}{a} \]

      4. associate--l+ 1.9%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \color{blue}{\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(-1 - 1\right)\right)}}{a} \]

      5. metadata-eval 1.9%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-2}\right)}{a} \]

   7. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + -2\right)}}{a} \]

   8. Taylor expanded in c around inf 13.9%

      \[\leadsto \color{blue}{c \cdot \left(2 \cdot \frac{b\_2}{a \cdot c} - 0.5 \cdot \frac{1}{b\_2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 13.9%

         \[\leadsto c \cdot \left(\color{blue}{\frac{2 \cdot b\_2}{a \cdot c}} - 0.5 \cdot \frac{1}{b\_2}\right) \]

      2. associate-*r/ 13.9%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \color{blue}{\frac{0.5 \cdot 1}{b\_2}}\right) \]

      3. metadata-eval 13.9%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{\color{blue}{0.5}}{b\_2}\right) \]

   10. Simplified 13.9%

       \[\leadsto \color{blue}{c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{0.5}{b\_2}\right)} \]

   if -9.50000000000000015e131 < b_2 < 2.5e-80

   1. Initial program 50.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 50.4%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]

      2. *-commutative 50.4%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      3. fmm-def 50.4%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      4. prod-diff 50.4%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b\_2, b\_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      5. *-commutative 50.4%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      6. fmm-def 50.4%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]

      7. associate-+l+ 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]

      8. sub-neg 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(b\_2 \cdot b\_2 + \left(-a \cdot c\right)\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      9. +-commutative 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(\left(-a \cdot c\right) + b\_2 \cdot b\_2\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      10. distribute-rgt-neg-in 50.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\color{blue}{a \cdot \left(-c\right)} + b\_2 \cdot b\_2\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      11. fma-define 50.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      12. pow2 50.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, \color{blue}{{b\_2}^{2}}\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      13. *-commutative 50.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      14. fma-undefine 50.4%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      15. distribute-lft-neg-in 50.4%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      16. *-commutative 50.4%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      17. distribute-rgt-neg-in 50.4%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      18. fma-define 50.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]

      19. *-commutative 50.3%

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]

   4. Applied egg-rr 50.3%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, {b\_2}^{2}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   5. Step-by-step derivation

      1. fma-define 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(a \cdot \left(-c\right) + {b\_2}^{2}\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      2. +-commutative 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} + a \cdot \left(-c\right)\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      3. *-commutative 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} + \color{blue}{\left(-c\right) \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      4. cancel-sign-sub-inv 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - c \cdot a\right)} + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]

      5. count-2 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

      6. *-commutative 50.3%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left({b\_2}^{2} - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   6. Simplified 50.3%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left({b\_2}^{2} - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]

   7. Taylor expanded in c around inf 21.3%

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(\frac{1}{a} \cdot \sqrt{\frac{2 \cdot \left(a + -1 \cdot a\right) - a}{c}}\right) + -1 \cdot \frac{b\_2}{a \cdot c}\right)} \]

   8. Taylor expanded in c around 0 45.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{a} \cdot \sqrt{c \cdot \left(2 \cdot \left(a + -1 \cdot a\right) - a\right)}\right) + -1 \cdot \frac{b\_2}{a}} \]
   9. Step-by-step derivation

      1. +-commutative 45.8%

         \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} + -1 \cdot \left(\frac{1}{a} \cdot \sqrt{c \cdot \left(2 \cdot \left(a + -1 \cdot a\right) - a\right)}\right)} \]

      2. mul-1-neg 45.8%

         \[\leadsto -1 \cdot \frac{b\_2}{a} + \color{blue}{\left(-\frac{1}{a} \cdot \sqrt{c \cdot \left(2 \cdot \left(a + -1 \cdot a\right) - a\right)}\right)} \]

      3. unsub-neg 45.8%

         \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} - \frac{1}{a} \cdot \sqrt{c \cdot \left(2 \cdot \left(a + -1 \cdot a\right) - a\right)}} \]

      4. associate-*r/ 45.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot b\_2}{a}} - \frac{1}{a} \cdot \sqrt{c \cdot \left(2 \cdot \left(a + -1 \cdot a\right) - a\right)} \]

      5. mul-1-neg 45.8%

         \[\leadsto \frac{\color{blue}{-b\_2}}{a} - \frac{1}{a} \cdot \sqrt{c \cdot \left(2 \cdot \left(a + -1 \cdot a\right) - a\right)} \]

      6. associate-*l/ 45.9%

         \[\leadsto \frac{-b\_2}{a} - \color{blue}{\frac{1 \cdot \sqrt{c \cdot \left(2 \cdot \left(a + -1 \cdot a\right) - a\right)}}{a}} \]

      7. *-lft-identity 45.9%

         \[\leadsto \frac{-b\_2}{a} - \frac{\color{blue}{\sqrt{c \cdot \left(2 \cdot \left(a + -1 \cdot a\right) - a\right)}}}{a} \]

      8. distribute-rgt1-in 45.9%

         \[\leadsto \frac{-b\_2}{a} - \frac{\sqrt{c \cdot \left(2 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot a\right)} - a\right)}}{a} \]

      9. metadata-eval 45.9%

         \[\leadsto \frac{-b\_2}{a} - \frac{\sqrt{c \cdot \left(2 \cdot \left(\color{blue}{0} \cdot a\right) - a\right)}}{a} \]

      10. mul0-lft 45.9%

          \[\leadsto \frac{-b\_2}{a} - \frac{\sqrt{c \cdot \left(2 \cdot \color{blue}{0} - a\right)}}{a} \]

      11. metadata-eval 45.9%

          \[\leadsto \frac{-b\_2}{a} - \frac{\sqrt{c \cdot \left(\color{blue}{0} - a\right)}}{a} \]

      12. neg-sub0 45.9%

          \[\leadsto \frac{-b\_2}{a} - \frac{\sqrt{c \cdot \color{blue}{\left(-a\right)}}}{a} \]

   10. Simplified 45.9%

       \[\leadsto \color{blue}{\frac{-b\_2}{a} - \frac{\sqrt{c \cdot \left(-a\right)}}{a}} \]

   if 2.5e-80 < b_2

   1. Initial program 71.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 89.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{+131}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 2.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b\_2}{-a} - \frac{\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \left(c \cdot \frac{1}{b\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (* c (- (/ (* b_2 2.0) (* c a)) (/ 0.5 b_2)))
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (* c (/ 1.0 b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c * (1.0 / b_2)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = c * (((b_2 * 2.0d0) / (c * a)) - (0.5d0 / b_2))
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c * (1.0d0 / b_2)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c * (1.0 / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2))
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c * (1.0 / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(c * Float64(Float64(Float64(b_2 * 2.0) / Float64(c * a)) - Float64(0.5 / b_2)));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c * Float64(1.0 / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c * (1.0 / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(c * N[(N[(N[(b$95$2 * 2.0), $MachinePrecision] / N[(c * a), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c * N[(1.0 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 29.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 27.8%

         \[\leadsto \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. sqrt-unprod 29.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. sqr-neg 29.0%

         \[\leadsto \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. sqrt-prod 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. add-sqr-sqrt 25.9%

         \[\leadsto \frac{\color{blue}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. add-cube-cbrt 25.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right) \cdot \sqrt[3]{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. pow3 25.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 25.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   5. Taylor expanded in b_2 around -inf 3.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 3.1%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}}{a} \]

      2. neg-mul-1 3.1%

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right)} \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}{a} \]

      3. rem-cube-cbrt 3.1%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-1}\right) - 1\right)}{a} \]

      4. associate--l+ 3.1%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \color{blue}{\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(-1 - 1\right)\right)}}{a} \]

      5. metadata-eval 3.1%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-2}\right)}{a} \]

   7. Simplified 3.1%

      \[\leadsto \frac{\color{blue}{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + -2\right)}}{a} \]

   8. Taylor expanded in c around inf 7.3%

      \[\leadsto \color{blue}{c \cdot \left(2 \cdot \frac{b\_2}{a \cdot c} - 0.5 \cdot \frac{1}{b\_2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 7.3%

         \[\leadsto c \cdot \left(\color{blue}{\frac{2 \cdot b\_2}{a \cdot c}} - 0.5 \cdot \frac{1}{b\_2}\right) \]

      2. associate-*r/ 7.3%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \color{blue}{\frac{0.5 \cdot 1}{b\_2}}\right) \]

      3. metadata-eval 7.3%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{\color{blue}{0.5}}{b\_2}\right) \]

   10. Simplified 7.3%

       \[\leadsto \color{blue}{c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{0.5}{b\_2}\right)} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 74.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 72.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. div-inv 72.6%

         \[\leadsto -2 \cdot \frac{b\_2}{a} + 0.5 \cdot \color{blue}{\left(c \cdot \frac{1}{b\_2}\right)} \]

   5. Applied egg-rr 72.6%

      \[\leadsto -2 \cdot \frac{b\_2}{a} + 0.5 \cdot \color{blue}{\left(c \cdot \frac{1}{b\_2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \left(c \cdot \frac{1}{b\_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (* c (- (/ (* b_2 2.0) (* c a)) (/ 0.5 b_2)))
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = c * (((b_2 * 2.0d0) / (c * a)) - (0.5d0 / b_2))
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2))
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(c * Float64(Float64(Float64(b_2 * 2.0) / Float64(c * a)) - Float64(0.5 / b_2)));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = c * (((b_2 * 2.0) / (c * a)) - (0.5 / b_2));
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(c * N[(N[(N[(b$95$2 * 2.0), $MachinePrecision] / N[(c * a), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 29.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 27.8%

         \[\leadsto \frac{\color{blue}{\sqrt{-b\_2} \cdot \sqrt{-b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. sqrt-unprod 29.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\_2\right) \cdot \left(-b\_2\right)}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. sqr-neg 29.0%

         \[\leadsto \frac{\sqrt{\color{blue}{b\_2 \cdot b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      4. sqrt-prod 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2} \cdot \sqrt{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. add-sqr-sqrt 25.9%

         \[\leadsto \frac{\color{blue}{b\_2} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      6. add-cube-cbrt 25.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right) \cdot \sqrt[3]{b\_2}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      7. pow3 25.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 25.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   5. Taylor expanded in b_2 around -inf 3.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b\_2 \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)\right)}}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 3.1%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}}{a} \]

      2. neg-mul-1 3.1%

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right)} \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)}{a} \]

      3. rem-cube-cbrt 3.1%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-1}\right) - 1\right)}{a} \]

      4. associate--l+ 3.1%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \color{blue}{\left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(-1 - 1\right)\right)}}{a} \]

      5. metadata-eval 3.1%

         \[\leadsto \frac{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + \color{blue}{-2}\right)}{a} \]

   7. Simplified 3.1%

      \[\leadsto \frac{\color{blue}{\left(-b\_2\right) \cdot \left(0.5 \cdot \frac{a \cdot c}{{b\_2}^{2}} + -2\right)}}{a} \]

   8. Taylor expanded in c around inf 7.3%

      \[\leadsto \color{blue}{c \cdot \left(2 \cdot \frac{b\_2}{a \cdot c} - 0.5 \cdot \frac{1}{b\_2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 7.3%

         \[\leadsto c \cdot \left(\color{blue}{\frac{2 \cdot b\_2}{a \cdot c}} - 0.5 \cdot \frac{1}{b\_2}\right) \]

      2. associate-*r/ 7.3%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \color{blue}{\frac{0.5 \cdot 1}{b\_2}}\right) \]

      3. metadata-eval 7.3%

         \[\leadsto c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{\color{blue}{0.5}}{b\_2}\right) \]

   10. Simplified 7.3%

       \[\leadsto \color{blue}{c \cdot \left(\frac{2 \cdot b\_2}{a \cdot c} - \frac{0.5}{b\_2}\right)} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 74.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 72.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c \cdot \left(\frac{b\_2 \cdot 2}{c \cdot a} - \frac{0.5}{b\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	return (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
end function

Java

public static double code(double a, double b_2, double c) {
	return (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
}

Python

def code(a, b_2, c):
	return (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))

Julia

function code(a, b_2, c)
	return Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.9%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in c around 0 32.4%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
4. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{c}{t\_1 - b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + t\_1}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024179 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))