quadm (p42, negative)

Percentage Accurate: 51.6% → 87.6%

Time: 13.0s

Alternatives: 8

Speedup: 12.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+26}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-181}:\\ \;\;\;\;\frac{c \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{-a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+26)
   (/ (- c) b)
   (if (<= b -4.2e-181)
     (/ (* c -2.0) (- b (hypot b (sqrt (* a (* c -4.0))))))
     (if (<= b 4.2e+109)
       (/ (+ b (sqrt (- (* b b) (* 4.0 (* c a))))) (- (* a 2.0)))
       (- (/ c b) (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+26) {
		tmp = -c / b;
	} else if (b <= -4.2e-181) {
		tmp = (c * -2.0) / (b - hypot(b, sqrt((a * (c * -4.0)))));
	} else if (b <= 4.2e+109) {
		tmp = (b + sqrt(((b * b) - (4.0 * (c * a))))) / -(a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+26) {
		tmp = -c / b;
	} else if (b <= -4.2e-181) {
		tmp = (c * -2.0) / (b - Math.hypot(b, Math.sqrt((a * (c * -4.0)))));
	} else if (b <= 4.2e+109) {
		tmp = (b + Math.sqrt(((b * b) - (4.0 * (c * a))))) / -(a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+26:
		tmp = -c / b
	elif b <= -4.2e-181:
		tmp = (c * -2.0) / (b - math.hypot(b, math.sqrt((a * (c * -4.0)))))
	elif b <= 4.2e+109:
		tmp = (b + math.sqrt(((b * b) - (4.0 * (c * a))))) / -(a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+26)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -4.2e-181)
		tmp = Float64(Float64(c * -2.0) / Float64(b - hypot(b, sqrt(Float64(a * Float64(c * -4.0))))));
	elseif (b <= 4.2e+109)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(-Float64(a * 2.0)));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+26)
		tmp = -c / b;
	elseif (b <= -4.2e-181)
		tmp = (c * -2.0) / (b - hypot(b, sqrt((a * (c * -4.0)))));
	elseif (b <= 4.2e+109)
		tmp = (b + sqrt(((b * b) - (4.0 * (c * a))))) / -(a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+26], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -4.2e-181], N[(N[(c * -2.0), $MachinePrecision] / N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.2e+109], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(a * 2.0), $MachinePrecision])), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0000000000000001e26

   1. Initial program 7.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 5.5%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 5.5%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 5.5%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 5.5%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 3.0%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 3.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 3.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 3.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 5.5%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 7.4%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 7.4%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 7.4%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 7.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 7.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 7.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 7.4%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 90.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 90.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 90.1%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   7. Simplified 90.1%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -2.0000000000000001e26 < b < -4.20000000000000006e-181

   1. Initial program 59.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 59.0%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 59.0%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 59.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 59.0%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 58.8%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 58.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 58.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 58.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 59.1%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 59.1%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 59.1%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 59.1%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 59.2%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 59.2%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 59.2%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 59.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{-0.5}}} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \]

      2. flip-+ 58.8%

         \[\leadsto \frac{1}{\frac{a}{-0.5}} \cdot \color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}} \]

      3. frac-times 49.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(b \cdot b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}{\frac{a}{-0.5} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}} \]

   6. Applied egg-rr 49.4%

      \[\leadsto \color{blue}{\frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* 58.6%

         \[\leadsto \color{blue}{\frac{\frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{a \cdot -2}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}} \]

      2. *-lft-identity 58.6%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left({b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)\right)}}{a \cdot -2}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      3. *-commutative 58.6%

         \[\leadsto \frac{\frac{1 \cdot \left({b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)\right)}{\color{blue}{-2 \cdot a}}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      4. times-frac 58.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{-2} \cdot \frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{a}}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      5. metadata-eval 58.6%

         \[\leadsto \frac{\color{blue}{-0.5} \cdot \frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{a}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      6. metadata-eval 58.6%

         \[\leadsto \frac{\color{blue}{\frac{0.5}{-1}} \cdot \frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{a}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      7. times-frac 58.6%

         \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot \left({b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)\right)}{-1 \cdot a}}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      8. neg-mul-1 58.6%

         \[\leadsto \frac{\frac{0.5 \cdot \left({b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)\right)}{\color{blue}{-a}}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      9. associate-/r* 49.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left({b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)\right)}{\left(-a\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)}} \]

      10. times-frac 58.6%

          \[\leadsto \color{blue}{\frac{0.5}{-a} \cdot \frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}} \]

      11. distribute-frac-neg2 58.6%

          \[\leadsto \color{blue}{\left(-\frac{0.5}{a}\right)} \cdot \frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      12. distribute-neg-frac 58.6%

          \[\leadsto \color{blue}{\frac{-0.5}{a}} \cdot \frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      13. metadata-eval 58.6%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \frac{{b}^{2} - \left({b}^{2} + c \cdot \left(a \cdot -4\right)\right)}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

   8. Simplified 58.0%

      \[\leadsto \color{blue}{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{\frac{-0.5}{a}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 58.0%

         \[\leadsto \color{blue}{\left(\left(c \cdot a\right) \cdot 4\right)} \cdot \frac{\frac{-0.5}{a}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

      2. associate-*l* 58.0%

         \[\leadsto \color{blue}{\left(c \cdot a\right) \cdot \left(4 \cdot \frac{\frac{-0.5}{a}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\right)} \]

      3. associate-/l/ 57.9%

         \[\leadsto \left(c \cdot a\right) \cdot \left(4 \cdot \color{blue}{\frac{-0.5}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a}}\right) \]

   10. Applied egg-rr 57.9%

       \[\leadsto \color{blue}{\left(c \cdot a\right) \cdot \left(4 \cdot \frac{-0.5}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 57.9%

          \[\leadsto \color{blue}{\left(a \cdot c\right)} \cdot \left(4 \cdot \frac{-0.5}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a}\right) \]

       2. associate-*r/ 57.9%

          \[\leadsto \left(a \cdot c\right) \cdot \color{blue}{\frac{4 \cdot -0.5}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a}} \]

       3. metadata-eval 57.9%

          \[\leadsto \left(a \cdot c\right) \cdot \frac{\color{blue}{-2}}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right) \cdot a} \]

       4. *-commutative 57.9%

          \[\leadsto \left(a \cdot c\right) \cdot \frac{-2}{\color{blue}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)}} \]

       5. associate-*r* 57.9%

          \[\leadsto \left(a \cdot c\right) \cdot \frac{-2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4}}\right)\right)} \]

       6. *-commutative 57.9%

          \[\leadsto \left(a \cdot c\right) \cdot \frac{-2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}\right)\right)} \]

       7. rem-square-sqrt 0.0%

          \[\leadsto \left(a \cdot c\right) \cdot \frac{-2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)\right)} \]

       8. unpow2 0.0%

          \[\leadsto \left(a \cdot c\right) \cdot \frac{-2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot \color{blue}{{\left(\sqrt{-4}\right)}^{2}}}\right)\right)} \]

       9. associate-*r* 0.0%

          \[\leadsto \left(a \cdot c\right) \cdot \frac{-2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\right)\right)} \]

       10. unpow2 0.0%

           \[\leadsto \left(a \cdot c\right) \cdot \frac{-2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}\right)}\right)\right)} \]

       11. rem-square-sqrt 57.9%

           \[\leadsto \left(a \cdot c\right) \cdot \frac{-2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot \color{blue}{-4}\right)}\right)\right)} \]

   12. Simplified 57.9%

       \[\leadsto \color{blue}{\left(a \cdot c\right) \cdot \frac{-2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-*r/ 58.1%

          \[\leadsto \color{blue}{\frac{\left(a \cdot c\right) \cdot -2}{a \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}} \]

       2. *-commutative 58.1%

          \[\leadsto \frac{\left(a \cdot c\right) \cdot -2}{\color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot a}} \]

       3. times-frac 67.1%

          \[\leadsto \color{blue}{\frac{a \cdot c}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{-2}{a}} \]

       4. *-commutative 67.1%

          \[\leadsto \frac{\color{blue}{c \cdot a}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot \frac{-2}{a} \]

       5. associate-*r* 67.1%

          \[\leadsto \frac{c \cdot a}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} \cdot \frac{-2}{a} \]

       6. *-commutative 67.1%

          \[\leadsto \frac{c \cdot a}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}\right)} \cdot \frac{-2}{a} \]

       7. associate-*r* 67.1%

          \[\leadsto \frac{c \cdot a}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right)} \cdot \frac{-2}{a} \]

   14. Applied egg-rr 67.1%

       \[\leadsto \color{blue}{\frac{c \cdot a}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{-2}{a}} \]
   15. Step-by-step derivation

       1. associate-*l/ 67.2%

          \[\leadsto \color{blue}{\frac{\left(c \cdot a\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}} \]

       2. *-commutative 67.2%

          \[\leadsto \frac{\color{blue}{\left(a \cdot c\right)} \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \]

       3. associate-*r* 67.2%

          \[\leadsto \frac{\left(a \cdot c\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4}}\right)} \]

       4. *-commutative 67.2%

          \[\leadsto \frac{\left(a \cdot c\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}\right)} \]

       5. rem-square-sqrt 0.0%

          \[\leadsto \frac{\left(a \cdot c\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)} \]

       6. unpow2 0.0%

          \[\leadsto \frac{\left(a \cdot c\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot \color{blue}{{\left(\sqrt{-4}\right)}^{2}}}\right)} \]

       7. associate-*r* 0.0%

          \[\leadsto \frac{\left(a \cdot c\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\right)} \]

       8. unpow2 0.0%

          \[\leadsto \frac{\left(a \cdot c\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}\right)}\right)} \]

       9. rem-square-sqrt 67.2%

          \[\leadsto \frac{\left(a \cdot c\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot \color{blue}{-4}\right)}\right)} \]

   16. Simplified 67.2%

       \[\leadsto \color{blue}{\frac{\left(a \cdot c\right) \cdot \frac{-2}{a}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} \]

   17. Taylor expanded in a around 0 85.0%

       \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

   if -4.20000000000000006e-181 < b < 4.2000000000000003e109

   1. Initial program 89.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   if 4.2000000000000003e109 < b

   1. Initial program 56.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 56.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 56.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 56.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 56.6%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 56.5%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 56.5%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 56.5%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 56.5%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 56.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 56.4%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 56.4%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 56.4%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 56.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 56.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 56.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0 94.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 94.3%

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 94.3%

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

      3. unsub-neg 94.3%

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

   7. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+26}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-181}:\\ \;\;\;\;\frac{c \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{-a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{if}\;b \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-58}:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{1}{b} + a \cdot \frac{c}{{b}^{3}}\right) - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0)))))))
   (if (<= b -2.9e-5)
     (/ (- c) b)
     (if (<= b 4e-127)
       t_0
       (if (<= b 1.02e-58)
         (/ (- (* a (/ c b)) b) a)
         (if (<= b 1.6e-13)
           t_0
           (- (* c (+ (/ 1.0 b) (* a (/ c (pow b 3.0))))) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	double tmp;
	if (b <= -2.9e-5) {
		tmp = -c / b;
	} else if (b <= 4e-127) {
		tmp = t_0;
	} else if (b <= 1.02e-58) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 1.6e-13) {
		tmp = t_0;
	} else {
		tmp = (c * ((1.0 / b) + (a * (c / pow(b, 3.0))))) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
    if (b <= (-2.9d-5)) then
        tmp = -c / b
    else if (b <= 4d-127) then
        tmp = t_0
    else if (b <= 1.02d-58) then
        tmp = ((a * (c / b)) - b) / a
    else if (b <= 1.6d-13) then
        tmp = t_0
    else
        tmp = (c * ((1.0d0 / b) + (a * (c / (b ** 3.0d0))))) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	double tmp;
	if (b <= -2.9e-5) {
		tmp = -c / b;
	} else if (b <= 4e-127) {
		tmp = t_0;
	} else if (b <= 1.02e-58) {
		tmp = ((a * (c / b)) - b) / a;
	} else if (b <= 1.6e-13) {
		tmp = t_0;
	} else {
		tmp = (c * ((1.0 / b) + (a * (c / Math.pow(b, 3.0))))) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	tmp = 0
	if b <= -2.9e-5:
		tmp = -c / b
	elif b <= 4e-127:
		tmp = t_0
	elif b <= 1.02e-58:
		tmp = ((a * (c / b)) - b) / a
	elif b <= 1.6e-13:
		tmp = t_0
	else:
		tmp = (c * ((1.0 / b) + (a * (c / math.pow(b, 3.0))))) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))))
	tmp = 0.0
	if (b <= -2.9e-5)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 4e-127)
		tmp = t_0;
	elseif (b <= 1.02e-58)
		tmp = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
	elseif (b <= 1.6e-13)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * Float64(Float64(1.0 / b) + Float64(a * Float64(c / (b ^ 3.0))))) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	tmp = 0.0;
	if (b <= -2.9e-5)
		tmp = -c / b;
	elseif (b <= 4e-127)
		tmp = t_0;
	elseif (b <= 1.02e-58)
		tmp = ((a * (c / b)) - b) / a;
	elseif (b <= 1.6e-13)
		tmp = t_0;
	else
		tmp = (c * ((1.0 / b) + (a * (c / (b ^ 3.0))))) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.9e-5], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 4e-127], t$95$0, If[LessEqual[b, 1.02e-58], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.6e-13], t$95$0, N[(N[(c * N[(N[(1.0 / b), $MachinePrecision] + N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.9e-5

   1. Initial program 9.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 7.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 7.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 7.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 7.6%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 5.3%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 7.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 9.3%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 9.3%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 9.3%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 87.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 87.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 87.8%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   7. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -2.9e-5 < b < 4.0000000000000001e-127 or 1.0199999999999999e-58 < b < 1.6e-13

   1. Initial program 80.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 80.2%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 80.2%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 80.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 80.2%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 80.2%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 80.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 80.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 80.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 80.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 80.2%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 80.2%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 80.2%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 80.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 80.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 80.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 75.4%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 75.4%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}\right) \]

      2. *-commutative 75.4%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot -4\right)} \cdot c}\right) \]

      3. *-commutative 75.4%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right) \]

   7. Simplified 75.4%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right) \]

   if 4.0000000000000001e-127 < b < 1.0199999999999999e-58

   1. Initial program 94.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 94.3%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 94.3%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 94.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 94.3%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 94.2%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 94.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 94.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 94.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 94.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 94.0%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 94.0%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 94.0%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 94.0%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 94.0%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 94.0%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 66.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{a}} \]
   6. Step-by-step derivation

      1. +-commutative 66.7%

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} + -1 \cdot b}}{a} \]

      2. mul-1-neg 66.7%

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

      3. sub-neg 66.7%

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

      4. associate-/l* 66.7%

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

   7. Simplified 66.7%

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

   if 1.6e-13 < b

   1. Initial program 70.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 70.4%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 70.4%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 70.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 70.4%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 70.3%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 70.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 70.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 70.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 70.2%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 70.2%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 70.2%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 70.2%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 70.2%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 70.2%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 70.2%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0 87.1%

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

      1. +-commutative 87.1%

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

      2. mul-1-neg 87.1%

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

      3. unsub-neg 87.1%

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

      4. associate-/l* 90.0%

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

   7. Simplified 90.0%

      \[\leadsto \color{blue}{c \cdot \left(\frac{1}{b} + a \cdot \frac{c}{{b}^{3}}\right) - \frac{b}{a}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{-a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-5)
   (/ (- c) b)
   (if (<= b 8.5e+109)
     (/ (+ b (sqrt (- (* b b) (* 4.0 (* c a))))) (- (* a 2.0)))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-5) {
		tmp = -c / b;
	} else if (b <= 8.5e+109) {
		tmp = (b + sqrt(((b * b) - (4.0 * (c * a))))) / -(a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.4d-5)) then
        tmp = -c / b
    else if (b <= 8.5d+109) then
        tmp = (b + sqrt(((b * b) - (4.0d0 * (c * a))))) / -(a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-5) {
		tmp = -c / b;
	} else if (b <= 8.5e+109) {
		tmp = (b + Math.sqrt(((b * b) - (4.0 * (c * a))))) / -(a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e-5:
		tmp = -c / b
	elif b <= 8.5e+109:
		tmp = (b + math.sqrt(((b * b) - (4.0 * (c * a))))) / -(a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e-5)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 8.5e+109)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(-Float64(a * 2.0)));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.4e-5)
		tmp = -c / b;
	elseif (b <= 8.5e+109)
		tmp = (b + sqrt(((b * b) - (4.0 * (c * a))))) / -(a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e-5], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 8.5e+109], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(a * 2.0), $MachinePrecision])), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.3999999999999999e-5

   1. Initial program 9.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 7.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 7.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 7.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 7.6%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 5.3%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 7.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 9.3%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 9.3%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 9.3%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 87.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 87.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 87.8%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   7. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -4.3999999999999999e-5 < b < 8.5000000000000004e109

   1. Initial program 85.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   if 8.5000000000000004e109 < b

   1. Initial program 56.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 56.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 56.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 56.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 56.6%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 56.5%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 56.5%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 56.5%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 56.5%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 56.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 56.4%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 56.4%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 56.4%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 56.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 56.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 56.4%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0 94.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 94.3%

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 94.3%

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

      3. unsub-neg 94.3%

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

   7. Simplified 94.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{-a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-5)
   (/ (- c) b)
   (if (<= b 4e-127)
     (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-5) {
		tmp = -c / b;
	} else if (b <= 4e-127) {
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.9d-5)) then
        tmp = -c / b
    else if (b <= 4d-127) then
        tmp = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-5) {
		tmp = -c / b;
	} else if (b <= 4e-127) {
		tmp = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-5:
		tmp = -c / b
	elif b <= 4e-127:
		tmp = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-5)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 4e-127)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-5)
		tmp = -c / b;
	elseif (b <= 4e-127)
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-5], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 4e-127], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.9e-5

   1. Initial program 9.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 7.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 7.6%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 7.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 7.6%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 5.3%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 5.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 7.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 9.3%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 9.3%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 9.3%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 9.3%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 9.3%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 87.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 87.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 87.8%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   7. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -2.9e-5 < b < 4.0000000000000001e-127

   1. Initial program 77.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 77.9%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 77.9%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 77.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 77.9%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 77.9%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 77.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 77.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 77.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 78.0%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 78.0%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 78.0%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 78.0%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 78.0%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 78.0%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 78.0%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 75.2%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 75.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}\right) \]

      2. *-commutative 75.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\left(a \cdot -4\right)} \cdot c}\right) \]

      3. *-commutative 75.2%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right) \]

   7. Simplified 75.2%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right) \]

   if 4.0000000000000001e-127 < b

   1. Initial program 76.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 76.7%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 76.7%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 76.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 76.7%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 76.6%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 76.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 76.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 76.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 76.5%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 76.5%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 76.5%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 76.5%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 76.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 76.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 76.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 76.5%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0 81.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 81.2%

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 81.2%

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

      3. unsub-neg 81.2%

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

   7. Simplified 81.2%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-311)) then
        tmp = -c / b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 29.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 28.2%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 28.2%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 28.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 28.2%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 26.6%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 26.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 26.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 26.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 28.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 29.4%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 29.4%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 29.4%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 29.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 29.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 29.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 29.5%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 67.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 67.1%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   7. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -5.00000000000023e-311 < b

   1. Initial program 78.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 78.0%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 78.0%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 78.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 78.0%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 77.9%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 77.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 77.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 77.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 77.8%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 77.8%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 77.8%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 77.8%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 77.8%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 77.8%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 77.8%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 77.8%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0 65.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 65.5%

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 65.5%

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

      3. unsub-neg 65.5%

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

   7. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-279}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-279) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-279) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.7d-279)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-279) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-279:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-279)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.7e-279)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-279], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.70000000000000038e-279

   1. Initial program 28.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 27.3%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 27.3%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 27.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 27.3%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 25.6%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 25.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 25.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 25.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 27.3%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 28.5%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 28.5%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 28.5%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 28.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 28.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 28.5%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 28.5%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 68.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 68.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-1 68.7%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   7. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -3.70000000000000038e-279 < b

   1. Initial program 77.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 77.8%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 77.8%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 77.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 77.8%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 77.7%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 77.7%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 77.7%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 77.7%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 77.6%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 77.6%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 77.6%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 77.6%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 77.6%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 77.6%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 77.6%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 63.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 63.2%

         \[\leadsto \color{blue}{-\frac{b}{a}} \]

      2. distribute-neg-frac2 63.2%

         \[\leadsto \color{blue}{\frac{b}{-a}} \]

   7. Simplified 63.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-279}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -2.9e+91) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e+91) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.9d+91)) then
        tmp = c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e+91) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e+91:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e+91)
		tmp = Float64(c / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e+91)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e+91], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.90000000000000014e91

   1. Initial program 6.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 4.4%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 4.4%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 4.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 4.4%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 1.4%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 1.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 1.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 1.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 4.4%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 6.6%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 6.6%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 6.6%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 6.6%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 6.6%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 6.6%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 6.6%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 1.8%

      \[\leadsto \color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{a}} \]
   6. Step-by-step derivation

      1. +-commutative 1.8%

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} + -1 \cdot b}}{a} \]

      2. mul-1-neg 1.8%

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

      3. sub-neg 1.8%

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

      4. associate-/l* 2.1%

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

   7. Simplified 2.1%

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

   8. Taylor expanded in a around inf 37.9%

      \[\leadsto \color{blue}{\frac{c}{b}} \]

   if -2.90000000000000014e91 < b

   1. Initial program 71.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. div-sub 71.0%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      2. sub-neg 71.0%

         \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

      3. neg-mul-1 71.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      4. *-commutative 71.0%

         \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      5. associate-/l* 70.9%

         \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

      6. distribute-neg-frac 70.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

      7. neg-mul-1 70.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

      8. *-commutative 70.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

      9. associate-/l* 70.9%

         \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

      10. distribute-rgt-out 70.9%

          \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 70.9%

          \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 70.9%

          \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      13. fma-neg 70.9%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

      14. *-commutative 70.9%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

      15. *-commutative 70.9%

          \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

   3. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 46.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. mul-1-neg 46.0%

         \[\leadsto \color{blue}{-\frac{b}{a}} \]

      2. distribute-neg-frac2 46.0%

         \[\leadsto \color{blue}{\frac{b}{-a}} \]

   7. Simplified 46.0%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+91}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.1% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

public static double code(double a, double b, double c) {
	return c / b;
}

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / b;
end

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. div-sub 54.1%

      \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   2. sub-neg 54.1%

      \[\leadsto \color{blue}{\frac{-b}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)} \]

   3. neg-mul-1 54.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot b}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

   4. *-commutative 54.1%

      \[\leadsto \frac{\color{blue}{b \cdot -1}}{2 \cdot a} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

   5. associate-/l* 53.3%

      \[\leadsto \color{blue}{b \cdot \frac{-1}{2 \cdot a}} + \left(-\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right) \]

   6. distribute-neg-frac 53.3%

      \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \]

   7. neg-mul-1 53.3%

      \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{-1 \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

   8. *-commutative 53.3%

      \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \frac{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot -1}}{2 \cdot a} \]

   9. associate-/l* 54.0%

      \[\leadsto b \cdot \frac{-1}{2 \cdot a} + \color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \frac{-1}{2 \cdot a}} \]

   10. distribute-rgt-out 54.6%

       \[\leadsto \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

   11. associate-/r* 54.6%

       \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

   12. metadata-eval 54.6%

       \[\leadsto \frac{\color{blue}{-0.5}}{a} \cdot \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

   13. fma-neg 54.6%

       \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\right) \]

   14. *-commutative 54.6%

       \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}\right) \]

   15. *-commutative 54.6%

       \[\leadsto \frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \]

3. Simplified 54.6%

   \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 34.2%

   \[\leadsto \color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{a}} \]
6. Step-by-step derivation

   1. +-commutative 34.2%

      \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} + -1 \cdot b}}{a} \]

   2. mul-1-neg 34.2%

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

   3. sub-neg 34.2%

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

   4. associate-/l* 35.1%

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

7. Simplified 35.1%

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

8. Taylor expanded in a around inf 12.2%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
9. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024095 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0))) (/ (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))