Quadratic roots, full range

Percentage Accurate: 52.5% → 87.5%

Time: 19.9s

Alternatives: 11

Speedup: 12.9×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.5% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{-2}{\frac{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+153)
   (- (/ c b) (/ b a))
   (if (<= b -7.2e-153)
     (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
     (if (<= b 5.6e-96)
       (/ -2.0 (/ (+ b (hypot (sqrt (* a (* c -4.0))) b)) c))
       (/ c (- b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7.2e-153) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else if (b <= 5.6e-96) {
		tmp = -2.0 / ((b + hypot(sqrt((a * (c * -4.0))), b)) / c);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -7.2e-153)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	elseif (b <= 5.6e-96)
		tmp = Float64(-2.0 / Float64(Float64(b + hypot(sqrt(Float64(a * Float64(c * -4.0))), b)) / c));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7.2e-153], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-96], N[(-2.0 / N[(N[(b + N[Sqrt[N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e153

   1. Initial program 39.4%

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

      1. *-commutative 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in b around -inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. distribute-rgt-neg-in 97.7%

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

      3. +-commutative 97.7%

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

      4. mul-1-neg 97.7%

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

      5. unsub-neg 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in a around inf 98.2%

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

   if -2e153 < b < -7.1999999999999995e-153

   1. Initial program 92.8%

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

      1. *-commutative 92.8%

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

   3. Simplified 92.8%

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

   if -7.1999999999999995e-153 < b < 5.60000000000000031e-96

   1. Initial program 75.2%

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

      1. *-commutative 75.2%

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

   3. Simplified 75.2%

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

   6. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 75.0%

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

      2. distribute-rgt-out-- 75.0%

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

   8. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.2%

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

      2. clear-num 75.0%

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

   10. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)}}} \]
   11. Step-by-step derivation

       1. flip-- 72.5%

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

       2. add-sqr-sqrt 72.3%

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

       3. fma-undefine 72.3%

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

       4. unpow2 72.3%

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

       5. associate--l+ 72.3%

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

       6. *-commutative 72.3%

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

       7. associate-*r* 72.3%

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

       8. +-inverses 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \frac{\left(a \cdot -4\right) \cdot c + \color{blue}{0}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} + b}}} \]

       9. +-commutative 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \frac{\left(a \cdot -4\right) \cdot c + 0}{\color{blue}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}}} \]

   12. Applied egg-rr 72.3%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\frac{\left(a \cdot -4\right) \cdot c + 0}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}}} \]
   13. Step-by-step derivation

       1. div-inv 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(\left(\left(a \cdot -4\right) \cdot c + 0\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}\right)}}} \]

       2. +-rgt-identity 72.3%

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

       3. associate-*l* 72.3%

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

       4. fma-undefine 72.3%

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

       5. *-commutative 72.3%

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

       6. associate-*l* 72.3%

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

       7. +-rgt-identity 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\color{blue}{\left(\left(a \cdot -4\right) \cdot c + 0\right)} + {b}^{2}}}\right)}} \]

       8. add-sqr-sqrt 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\color{blue}{\sqrt{\left(a \cdot -4\right) \cdot c + 0} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + 0}} + {b}^{2}}}\right)}} \]

       9. unpow2 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\sqrt{\left(a \cdot -4\right) \cdot c + 0} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + 0} + \color{blue}{b \cdot b}}}\right)}} \]

       10. hypot-define 72.0%

           \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \color{blue}{\mathsf{hypot}\left(\sqrt{\left(a \cdot -4\right) \cdot c + 0}, b\right)}}\right)}} \]

       11. +-rgt-identity 72.0%

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

       12. associate-*l* 72.0%

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

   14. Applied egg-rr 72.0%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(-4 \cdot c\right)}, b\right)}\right)}}} \]
   15. Step-by-step derivation

       1. associate-*r/ 72.0%

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

       2. *-rgt-identity 72.0%

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

       3. associate-/l* 76.3%

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

       4. *-commutative 76.3%

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

       5. *-commutative 76.3%

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

       6. *-commutative 76.3%

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

       7. associate-*l* 76.3%

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

   16. Simplified 76.3%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(a \cdot \frac{c \cdot -4}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right)}\right)}}} \]
   17. Step-by-step derivation

       1. div-inv 76.3%

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

       2. *-commutative 76.3%

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

       3. *-commutative 76.3%

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

       4. *-un-lft-identity 76.3%

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

       5. times-frac 76.3%

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

       6. metadata-eval 76.3%

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

       7. associate-*r* 76.3%

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

   18. Applied egg-rr 76.3%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{1}{\left(a \cdot \left(-4 \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{\left(c \cdot -4\right) \cdot a}, b\right)}\right)\right) \cdot 0.5}}} \]
   19. Step-by-step derivation

       1. associate-*r/ 76.3%

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

       2. times-frac 76.3%

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

       3. metadata-eval 76.3%

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

       4. *-commutative 76.3%

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

       5. associate-*r/ 76.3%

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

       6. *-commutative 76.3%

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

       7. times-frac 76.6%

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

       8. *-inverses 76.6%

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

       9. associate-*r/ 76.6%

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

       10. *-commutative 76.6%

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

       11. *-lft-identity 76.6%

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

       12. times-frac 76.5%

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

       13. /-rgt-identity 76.5%

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

       14. *-commutative 76.5%

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

   20. Simplified 76.5%

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \frac{2}{c \cdot \frac{-4}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}}}} \]
   21. Step-by-step derivation

       1. *-un-lft-identity 76.5%

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

       2. clear-num 76.5%

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

       3. div-inv 76.5%

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

       4. clear-num 76.5%

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

       5. un-div-inv 76.6%

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

       6. div-inv 76.6%

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

       7. metadata-eval 76.6%

          \[\leadsto \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot \color{blue}{-0.25}} \cdot \frac{1}{2} \]

       8. metadata-eval 76.6%

          \[\leadsto \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25} \cdot \color{blue}{0.5} \]

   22. Applied egg-rr 76.6%

       \[\leadsto \color{blue}{\frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25} \cdot 0.5} \]
   23. Step-by-step derivation

       1. *-commutative 76.6%

          \[\leadsto \color{blue}{0.5 \cdot \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25}} \]

       2. associate-*r/ 76.6%

          \[\leadsto \color{blue}{\frac{0.5 \cdot c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25}} \]

       3. *-commutative 76.6%

          \[\leadsto \frac{0.5 \cdot c}{\color{blue}{-0.25 \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)}} \]

       4. times-frac 76.6%

          \[\leadsto \color{blue}{\frac{0.5}{-0.25} \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}} \]

       5. metadata-eval 76.6%

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

   24. Simplified 76.6%

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

       1. clear-num 76.6%

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

       2. un-div-inv 76.6%

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

   26. Applied egg-rr 76.6%

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

   if 5.60000000000000031e-96 < b

   1. Initial program 12.3%

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

      1. *-commutative 12.3%

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

   3. Simplified 12.3%

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

   5. Taylor expanded in b around inf 90.4%

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

      1. associate-*r/ 90.4%

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

      2. mul-1-neg 90.4%

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

   7. Simplified 90.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{-2}{\frac{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{-2}{\frac{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+153)
   (- (/ c b) (/ b a))
   (if (<= b -7.2e-153)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (if (<= b 5.6e-96)
       (/ -2.0 (/ (+ b (hypot (sqrt (* a (* c -4.0))) b)) c))
       (/ c (- b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7.2e-153) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else if (b <= 5.6e-96) {
		tmp = -2.0 / ((b + hypot(sqrt((a * (c * -4.0))), b)) / c);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7.2e-153) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else if (b <= 5.6e-96) {
		tmp = -2.0 / ((b + Math.hypot(Math.sqrt((a * (c * -4.0))), b)) / c);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+153:
		tmp = (c / b) - (b / a)
	elif b <= -7.2e-153:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	elif b <= 5.6e-96:
		tmp = -2.0 / ((b + math.hypot(math.sqrt((a * (c * -4.0))), b)) / c)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -7.2e-153)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	elseif (b <= 5.6e-96)
		tmp = Float64(-2.0 / Float64(Float64(b + hypot(sqrt(Float64(a * Float64(c * -4.0))), b)) / c));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+153)
		tmp = (c / b) - (b / a);
	elseif (b <= -7.2e-153)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	elseif (b <= 5.6e-96)
		tmp = -2.0 / ((b + hypot(sqrt((a * (c * -4.0))), b)) / c);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7.2e-153], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-96], N[(-2.0 / N[(N[(b + N[Sqrt[N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e153

   1. Initial program 39.4%

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

      1. *-commutative 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in b around -inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. distribute-rgt-neg-in 97.7%

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

      3. +-commutative 97.7%

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

      4. mul-1-neg 97.7%

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

      5. unsub-neg 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in a around inf 98.2%

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

   if -2e153 < b < -7.1999999999999995e-153

   1. Initial program 92.8%

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

   if -7.1999999999999995e-153 < b < 5.60000000000000031e-96

   1. Initial program 75.2%

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

      1. *-commutative 75.2%

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

   3. Simplified 75.2%

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

   6. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 75.0%

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

      2. distribute-rgt-out-- 75.0%

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

   8. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.2%

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

      2. clear-num 75.0%

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

   10. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)}}} \]
   11. Step-by-step derivation

       1. flip-- 72.5%

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

       2. add-sqr-sqrt 72.3%

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

       3. fma-undefine 72.3%

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

       4. unpow2 72.3%

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

       5. associate--l+ 72.3%

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

       6. *-commutative 72.3%

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

       7. associate-*r* 72.3%

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

       8. +-inverses 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \frac{\left(a \cdot -4\right) \cdot c + \color{blue}{0}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} + b}}} \]

       9. +-commutative 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \frac{\left(a \cdot -4\right) \cdot c + 0}{\color{blue}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}}} \]

   12. Applied egg-rr 72.3%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\frac{\left(a \cdot -4\right) \cdot c + 0}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}}} \]
   13. Step-by-step derivation

       1. div-inv 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(\left(\left(a \cdot -4\right) \cdot c + 0\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}\right)}}} \]

       2. +-rgt-identity 72.3%

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

       3. associate-*l* 72.3%

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

       4. fma-undefine 72.3%

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

       5. *-commutative 72.3%

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

       6. associate-*l* 72.3%

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

       7. +-rgt-identity 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\color{blue}{\left(\left(a \cdot -4\right) \cdot c + 0\right)} + {b}^{2}}}\right)}} \]

       8. add-sqr-sqrt 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\color{blue}{\sqrt{\left(a \cdot -4\right) \cdot c + 0} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + 0}} + {b}^{2}}}\right)}} \]

       9. unpow2 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\sqrt{\left(a \cdot -4\right) \cdot c + 0} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + 0} + \color{blue}{b \cdot b}}}\right)}} \]

       10. hypot-define 72.0%

           \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \color{blue}{\mathsf{hypot}\left(\sqrt{\left(a \cdot -4\right) \cdot c + 0}, b\right)}}\right)}} \]

       11. +-rgt-identity 72.0%

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

       12. associate-*l* 72.0%

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

   14. Applied egg-rr 72.0%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(-4 \cdot c\right)}, b\right)}\right)}}} \]
   15. Step-by-step derivation

       1. associate-*r/ 72.0%

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

       2. *-rgt-identity 72.0%

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

       3. associate-/l* 76.3%

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

       4. *-commutative 76.3%

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

       5. *-commutative 76.3%

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

       6. *-commutative 76.3%

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

       7. associate-*l* 76.3%

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

   16. Simplified 76.3%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(a \cdot \frac{c \cdot -4}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right)}\right)}}} \]
   17. Step-by-step derivation

       1. div-inv 76.3%

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

       2. *-commutative 76.3%

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

       3. *-commutative 76.3%

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

       4. *-un-lft-identity 76.3%

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

       5. times-frac 76.3%

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

       6. metadata-eval 76.3%

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

       7. associate-*r* 76.3%

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

   18. Applied egg-rr 76.3%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{1}{\left(a \cdot \left(-4 \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{\left(c \cdot -4\right) \cdot a}, b\right)}\right)\right) \cdot 0.5}}} \]
   19. Step-by-step derivation

       1. associate-*r/ 76.3%

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

       2. times-frac 76.3%

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

       3. metadata-eval 76.3%

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

       4. *-commutative 76.3%

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

       5. associate-*r/ 76.3%

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

       6. *-commutative 76.3%

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

       7. times-frac 76.6%

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

       8. *-inverses 76.6%

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

       9. associate-*r/ 76.6%

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

       10. *-commutative 76.6%

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

       11. *-lft-identity 76.6%

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

       12. times-frac 76.5%

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

       13. /-rgt-identity 76.5%

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

       14. *-commutative 76.5%

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

   20. Simplified 76.5%

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \frac{2}{c \cdot \frac{-4}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}}}} \]
   21. Step-by-step derivation

       1. *-un-lft-identity 76.5%

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

       2. clear-num 76.5%

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

       3. div-inv 76.5%

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

       4. clear-num 76.5%

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

       5. un-div-inv 76.6%

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

       6. div-inv 76.6%

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

       7. metadata-eval 76.6%

          \[\leadsto \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot \color{blue}{-0.25}} \cdot \frac{1}{2} \]

       8. metadata-eval 76.6%

          \[\leadsto \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25} \cdot \color{blue}{0.5} \]

   22. Applied egg-rr 76.6%

       \[\leadsto \color{blue}{\frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25} \cdot 0.5} \]
   23. Step-by-step derivation

       1. *-commutative 76.6%

          \[\leadsto \color{blue}{0.5 \cdot \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25}} \]

       2. associate-*r/ 76.6%

          \[\leadsto \color{blue}{\frac{0.5 \cdot c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25}} \]

       3. *-commutative 76.6%

          \[\leadsto \frac{0.5 \cdot c}{\color{blue}{-0.25 \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)}} \]

       4. times-frac 76.6%

          \[\leadsto \color{blue}{\frac{0.5}{-0.25} \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}} \]

       5. metadata-eval 76.6%

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

   24. Simplified 76.6%

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

       1. clear-num 76.6%

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

       2. un-div-inv 76.6%

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

   26. Applied egg-rr 76.6%

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

   if 5.60000000000000031e-96 < b

   1. Initial program 12.3%

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

      1. *-commutative 12.3%

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

   3. Simplified 12.3%

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

   5. Taylor expanded in b around inf 90.4%

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

      1. associate-*r/ 90.4%

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

      2. mul-1-neg 90.4%

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

   7. Simplified 90.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{-2}{\frac{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-96}:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+153)
   (- (/ c b) (/ b a))
   (if (<= b -7.2e-153)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (if (<= b 5.6e-96)
       (* -2.0 (/ c (+ b (hypot (sqrt (* a (* c -4.0))) b))))
       (/ c (- b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7.2e-153) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else if (b <= 5.6e-96) {
		tmp = -2.0 * (c / (b + hypot(sqrt((a * (c * -4.0))), b)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7.2e-153) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else if (b <= 5.6e-96) {
		tmp = -2.0 * (c / (b + Math.hypot(Math.sqrt((a * (c * -4.0))), b)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+153:
		tmp = (c / b) - (b / a)
	elif b <= -7.2e-153:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	elif b <= 5.6e-96:
		tmp = -2.0 * (c / (b + math.hypot(math.sqrt((a * (c * -4.0))), b)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -7.2e-153)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	elseif (b <= 5.6e-96)
		tmp = Float64(-2.0 * Float64(c / Float64(b + hypot(sqrt(Float64(a * Float64(c * -4.0))), b))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+153)
		tmp = (c / b) - (b / a);
	elseif (b <= -7.2e-153)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	elseif (b <= 5.6e-96)
		tmp = -2.0 * (c / (b + hypot(sqrt((a * (c * -4.0))), b)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7.2e-153], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.6e-96], N[(-2.0 * N[(c / N[(b + N[Sqrt[N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e153

   1. Initial program 39.4%

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

      1. *-commutative 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in b around -inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. distribute-rgt-neg-in 97.7%

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

      3. +-commutative 97.7%

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

      4. mul-1-neg 97.7%

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

      5. unsub-neg 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in a around inf 98.2%

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

   if -2e153 < b < -7.1999999999999995e-153

   1. Initial program 92.8%

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

   if -7.1999999999999995e-153 < b < 5.60000000000000031e-96

   1. Initial program 75.2%

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

      1. *-commutative 75.2%

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

   3. Simplified 75.2%

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

   6. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 75.0%

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

      2. distribute-rgt-out-- 75.0%

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

   8. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)} \]
   9. Step-by-step derivation

      1. associate-*l/ 75.2%

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

      2. clear-num 75.0%

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

   10. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - b\right)}}} \]
   11. Step-by-step derivation

       1. flip-- 72.5%

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

       2. add-sqr-sqrt 72.3%

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

       3. fma-undefine 72.3%

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

       4. unpow2 72.3%

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

       5. associate--l+ 72.3%

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

       6. *-commutative 72.3%

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

       7. associate-*r* 72.3%

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

       8. +-inverses 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \frac{\left(a \cdot -4\right) \cdot c + \color{blue}{0}}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} + b}}} \]

       9. +-commutative 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \frac{\left(a \cdot -4\right) \cdot c + 0}{\color{blue}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}}} \]

   12. Applied egg-rr 72.3%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\frac{\left(a \cdot -4\right) \cdot c + 0}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}}} \]
   13. Step-by-step derivation

       1. div-inv 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(\left(\left(a \cdot -4\right) \cdot c + 0\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}\right)}}} \]

       2. +-rgt-identity 72.3%

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

       3. associate-*l* 72.3%

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

       4. fma-undefine 72.3%

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

       5. *-commutative 72.3%

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

       6. associate-*l* 72.3%

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

       7. +-rgt-identity 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\color{blue}{\left(\left(a \cdot -4\right) \cdot c + 0\right)} + {b}^{2}}}\right)}} \]

       8. add-sqr-sqrt 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\color{blue}{\sqrt{\left(a \cdot -4\right) \cdot c + 0} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + 0}} + {b}^{2}}}\right)}} \]

       9. unpow2 72.3%

          \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \sqrt{\sqrt{\left(a \cdot -4\right) \cdot c + 0} \cdot \sqrt{\left(a \cdot -4\right) \cdot c + 0} + \color{blue}{b \cdot b}}}\right)}} \]

       10. hypot-define 72.0%

           \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \color{blue}{\mathsf{hypot}\left(\sqrt{\left(a \cdot -4\right) \cdot c + 0}, b\right)}}\right)}} \]

       11. +-rgt-identity 72.0%

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

       12. associate-*l* 72.0%

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

   14. Applied egg-rr 72.0%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(\left(a \cdot \left(-4 \cdot c\right)\right) \cdot \frac{1}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(-4 \cdot c\right)}, b\right)}\right)}}} \]
   15. Step-by-step derivation

       1. associate-*r/ 72.0%

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

       2. *-rgt-identity 72.0%

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

       3. associate-/l* 76.3%

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

       4. *-commutative 76.3%

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

       5. *-commutative 76.3%

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

       6. *-commutative 76.3%

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

       7. associate-*l* 76.3%

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

   16. Simplified 76.3%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \color{blue}{\left(a \cdot \frac{c \cdot -4}{b + \mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right)}\right)}}} \]
   17. Step-by-step derivation

       1. div-inv 76.3%

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

       2. *-commutative 76.3%

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

       3. *-commutative 76.3%

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

       4. *-un-lft-identity 76.3%

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

       5. times-frac 76.3%

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

       6. metadata-eval 76.3%

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

       7. associate-*r* 76.3%

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

   18. Applied egg-rr 76.3%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{1}{\left(a \cdot \left(-4 \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{\left(c \cdot -4\right) \cdot a}, b\right)}\right)\right) \cdot 0.5}}} \]
   19. Step-by-step derivation

       1. associate-*r/ 76.3%

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

       2. times-frac 76.3%

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

       3. metadata-eval 76.3%

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

       4. *-commutative 76.3%

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

       5. associate-*r/ 76.3%

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

       6. *-commutative 76.3%

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

       7. times-frac 76.6%

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

       8. *-inverses 76.6%

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

       9. associate-*r/ 76.6%

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

       10. *-commutative 76.6%

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

       11. *-lft-identity 76.6%

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

       12. times-frac 76.5%

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

       13. /-rgt-identity 76.5%

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

       14. *-commutative 76.5%

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

   20. Simplified 76.5%

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \frac{2}{c \cdot \frac{-4}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}}}} \]
   21. Step-by-step derivation

       1. *-un-lft-identity 76.5%

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

       2. clear-num 76.5%

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

       3. div-inv 76.5%

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

       4. clear-num 76.5%

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

       5. un-div-inv 76.6%

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

       6. div-inv 76.6%

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

       7. metadata-eval 76.6%

          \[\leadsto \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot \color{blue}{-0.25}} \cdot \frac{1}{2} \]

       8. metadata-eval 76.6%

          \[\leadsto \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25} \cdot \color{blue}{0.5} \]

   22. Applied egg-rr 76.6%

       \[\leadsto \color{blue}{\frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25} \cdot 0.5} \]
   23. Step-by-step derivation

       1. *-commutative 76.6%

          \[\leadsto \color{blue}{0.5 \cdot \frac{c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25}} \]

       2. associate-*r/ 76.6%

          \[\leadsto \color{blue}{\frac{0.5 \cdot c}{\left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot -0.25}} \]

       3. *-commutative 76.6%

          \[\leadsto \frac{0.5 \cdot c}{\color{blue}{-0.25 \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)}} \]

       4. times-frac 76.6%

          \[\leadsto \color{blue}{\frac{0.5}{-0.25} \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}} \]

       5. metadata-eval 76.6%

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

   24. Simplified 76.6%

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

   if 5.60000000000000031e-96 < b

   1. Initial program 12.3%

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

      1. *-commutative 12.3%

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

   3. Simplified 12.3%

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

   5. Taylor expanded in b around inf 90.4%

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

      1. associate-*r/ 90.4%

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

      2. mul-1-neg 90.4%

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

   7. Simplified 90.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-96}:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2.85 \cdot 10^{-196}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))))
   (if (<= b -2e-8)
     (- (/ c b) (/ b a))
     (if (<= b -7e-192)
       t_0
       (if (<= b -2.85e-196) (/ b (- a)) (if (<= b 9e-92) t_0 (/ c (- b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -2e-8) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7e-192) {
		tmp = t_0;
	} else if (b <= -2.85e-196) {
		tmp = b / -a;
	} else if (b <= 9e-92) {
		tmp = t_0;
	} else {
		tmp = c / -b;
	}
	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 = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    if (b <= (-2d-8)) then
        tmp = (c / b) - (b / a)
    else if (b <= (-7d-192)) then
        tmp = t_0
    else if (b <= (-2.85d-196)) then
        tmp = b / -a
    else if (b <= 9d-92) then
        tmp = t_0
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	double tmp;
	if (b <= -2e-8) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7e-192) {
		tmp = t_0;
	} else if (b <= -2.85e-196) {
		tmp = b / -a;
	} else if (b <= 9e-92) {
		tmp = t_0;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	tmp = 0
	if b <= -2e-8:
		tmp = (c / b) - (b / a)
	elif b <= -7e-192:
		tmp = t_0
	elif b <= -2.85e-196:
		tmp = b / -a
	elif b <= 9e-92:
		tmp = t_0
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -2e-8)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -7e-192)
		tmp = t_0;
	elseif (b <= -2.85e-196)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 9e-92)
		tmp = t_0;
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	tmp = 0.0;
	if (b <= -2e-8)
		tmp = (c / b) - (b / a);
	elseif (b <= -7e-192)
		tmp = t_0;
	elseif (b <= -2.85e-196)
		tmp = b / -a;
	elseif (b <= 9e-92)
		tmp = t_0;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e-8], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7e-192], t$95$0, If[LessEqual[b, -2.85e-196], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 9e-92], t$95$0, N[(c / (-b)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e-8

   1. Initial program 61.2%

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

      1. *-commutative 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in b around -inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. distribute-rgt-neg-in 90.3%

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

      3. +-commutative 90.3%

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

      4. mul-1-neg 90.3%

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

      5. unsub-neg 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in a around inf 90.6%

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

   if -2e-8 < b < -7.00000000000000029e-192 or -2.8500000000000001e-196 < b < 9.0000000000000001e-92

   1. Initial program 84.6%

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

      1. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in b around 0 72.2%

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

      1. *-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-*r* 72.3%

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

   7. Simplified 72.3%

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

   if -7.00000000000000029e-192 < b < -2.8500000000000001e-196

   1. Initial program 10.8%

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

      1. *-commutative 10.8%

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

   3. Simplified 10.8%

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

   5. Taylor expanded in b around -inf 51.7%

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

      1. mul-1-neg 51.7%

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

      2. distribute-neg-frac2 51.7%

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

   7. Simplified 51.7%

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

   if 9.0000000000000001e-92 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.85 \cdot 10^{-196}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{if}\;b \leq -1.32 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2.85 \cdot 10^{-196}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 a) (- (sqrt (* a (* c -4.0))) b))))
   (if (<= b -1.32e-8)
     (- (/ c b) (/ b a))
     (if (<= b -7e-192)
       t_0
       (if (<= b -2.85e-196)
         (/ b (- a))
         (if (<= b 2.1e-91) t_0 (/ c (- b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	double tmp;
	if (b <= -1.32e-8) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7e-192) {
		tmp = t_0;
	} else if (b <= -2.85e-196) {
		tmp = b / -a;
	} else if (b <= 2.1e-91) {
		tmp = t_0;
	} else {
		tmp = c / -b;
	}
	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) * (sqrt((a * (c * (-4.0d0)))) - b)
    if (b <= (-1.32d-8)) then
        tmp = (c / b) - (b / a)
    else if (b <= (-7d-192)) then
        tmp = t_0
    else if (b <= (-2.85d-196)) then
        tmp = b / -a
    else if (b <= 2.1d-91) then
        tmp = t_0
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (0.5 / a) * (Math.sqrt((a * (c * -4.0))) - b);
	double tmp;
	if (b <= -1.32e-8) {
		tmp = (c / b) - (b / a);
	} else if (b <= -7e-192) {
		tmp = t_0;
	} else if (b <= -2.85e-196) {
		tmp = b / -a;
	} else if (b <= 2.1e-91) {
		tmp = t_0;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (0.5 / a) * (math.sqrt((a * (c * -4.0))) - b)
	tmp = 0
	if b <= -1.32e-8:
		tmp = (c / b) - (b / a)
	elif b <= -7e-192:
		tmp = t_0
	elif b <= -2.85e-196:
		tmp = b / -a
	elif b <= 2.1e-91:
		tmp = t_0
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(a * Float64(c * -4.0))) - b))
	tmp = 0.0
	if (b <= -1.32e-8)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= -7e-192)
		tmp = t_0;
	elseif (b <= -2.85e-196)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.1e-91)
		tmp = t_0;
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	tmp = 0.0;
	if (b <= -1.32e-8)
		tmp = (c / b) - (b / a);
	elseif (b <= -7e-192)
		tmp = t_0;
	elseif (b <= -2.85e-196)
		tmp = b / -a;
	elseif (b <= 2.1e-91)
		tmp = t_0;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.32e-8], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7e-192], t$95$0, If[LessEqual[b, -2.85e-196], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.1e-91], t$95$0, N[(c / (-b)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.32000000000000007e-8

   1. Initial program 61.2%

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

      1. *-commutative 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in b around -inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. distribute-rgt-neg-in 90.3%

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

      3. +-commutative 90.3%

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

      4. mul-1-neg 90.3%

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

      5. unsub-neg 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in a around inf 90.6%

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

   if -1.32000000000000007e-8 < b < -7.00000000000000029e-192 or -2.8500000000000001e-196 < b < 2.0999999999999999e-91

   1. Initial program 84.6%

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

      1. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   6. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} \cdot \frac{0.5}{a} + \left(-b \cdot \frac{0.5}{a}\right)} \]
   7. Step-by-step derivation

      1. sub-neg 84.4%

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

      2. distribute-rgt-out-- 84.4%

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

   8. Simplified 84.4%

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

   9. Taylor expanded in a around inf 72.0%

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

   11. Simplified 72.1%

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

   if -7.00000000000000029e-192 < b < -2.8500000000000001e-196

   1. Initial program 10.8%

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

      1. *-commutative 10.8%

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

   3. Simplified 10.8%

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

   5. Taylor expanded in b around -inf 51.7%

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

      1. mul-1-neg 51.7%

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

      2. distribute-neg-frac2 51.7%

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

   7. Simplified 51.7%

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

   if 2.0999999999999999e-91 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-8}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-192}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{elif}\;b \leq -2.85 \cdot 10^{-196}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+173}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.8e+173)
   (- (/ c b) (/ b a))
   (if (<= b 2.4e-91)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.8e+173) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.4e-91) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	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 <= (-8.8d+173)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.4d-91) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.8e+173) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.4e-91) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.8e+173:
		tmp = (c / b) - (b / a)
	elif b <= 2.4e-91:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.8e+173)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.4e-91)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.8e+173)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.4e-91)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.8e+173], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-91], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.7999999999999999e173

   1. Initial program 42.8%

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

      1. *-commutative 42.8%

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

   3. Simplified 42.8%

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

   5. Taylor expanded in b around -inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. distribute-rgt-neg-in 97.6%

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

      3. +-commutative 97.6%

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

      4. mul-1-neg 97.6%

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

      5. unsub-neg 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in a around inf 98.0%

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

   if -8.7999999999999999e173 < b < 2.40000000000000011e-91

   1. Initial program 82.7%

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

   if 2.40000000000000011e-91 < b

   1. Initial program 11.3%

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

      1. *-commutative 11.3%

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

   3. Simplified 11.3%

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

   5. Taylor expanded in b around inf 91.4%

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

      1. associate-*r/ 91.4%

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

      2. mul-1-neg 91.4%

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

   7. Simplified 91.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+173}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e-153) (- (/ c b) (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-153) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	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 <= (-7.2d-153)) then
        tmp = (c / b) - (b / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-153) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e-153:
		tmp = (c / b) - (b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e-153)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.2e-153)
		tmp = (c / b) - (b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e-153], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.1999999999999995e-153

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in b around -inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. distribute-rgt-neg-in 74.6%

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

      3. +-commutative 74.6%

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

      4. mul-1-neg 74.6%

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

      5. unsub-neg 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in a around inf 75.0%

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

   if -7.1999999999999995e-153 < b

   1. Initial program 36.1%

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

      1. *-commutative 36.1%

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

   3. Simplified 36.1%

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

   5. Taylor expanded in b around inf 58.7%

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

      1. associate-*r/ 58.7%

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

      2. mul-1-neg 58.7%

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

   7. Simplified 58.7%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.8e-232) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-232) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	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 <= (-6.8d-232)) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-232) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.8e-232:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.8e-232)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.8e-232)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.8e-232], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.8000000000000004e-232

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in b around -inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. distribute-neg-frac2 66.7%

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

   7. Simplified 66.7%

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

   if -6.8000000000000004e-232 < b

   1. Initial program 31.4%

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

      1. *-commutative 31.4%

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

   3. Simplified 31.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   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. mul-1-neg 67.1%

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

   7. Simplified 67.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-232}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -4.2e-199) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-199) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	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 <= (-4.2d-199)) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-199) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-199:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-199)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e-199)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-199], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.20000000000000004e-199

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in b around -inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. distribute-neg-frac2 69.3%

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

   7. Simplified 69.3%

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

   if -4.20000000000000004e-199 < b

   1. Initial program 34.5%

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

      1. *-commutative 34.5%

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

   3. Simplified 34.5%

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

   5. Taylor expanded in b around -inf 2.3%

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

      1. mul-1-neg 2.3%

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

      2. distribute-rgt-neg-in 2.3%

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

      3. +-commutative 2.3%

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

      4. mul-1-neg 2.3%

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

      5. unsub-neg 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in b around 0 14.2%

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

Alternative 10: 10.8% 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 52.0%

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

   1. *-commutative 52.0%

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

3. Simplified 52.0%

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

5. Taylor expanded in b around -inf 35.5%

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

   1. mul-1-neg 35.5%

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

   2. distribute-rgt-neg-in 35.5%

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

   3. +-commutative 35.5%

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

   4. mul-1-neg 35.5%

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

   5. unsub-neg 35.5%

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

7. Simplified 35.5%

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

8. Taylor expanded in b around 0 9.2%

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

Alternative 11: 5.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ c \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* c -0.5))

C

double code(double a, double b, double c) {
	return c * -0.5;
}

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 * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return c * -0.5;
}

Python

def code(a, b, c):
	return c * -0.5

Julia

function code(a, b, c)
	return Float64(c * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * -0.5;
end

Wolfram

code[a_, b_, c_] := N[(c * -0.5), $MachinePrecision]

Derivation

1. Initial program 52.0%

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

   1. *-commutative 52.0%

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

3. Simplified 52.0%

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

   1. *-un-lft-identity 52.0%

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

   2. *-un-lft-identity 52.0%

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

   3. prod-diff 52.0%

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

   4. *-commutative 52.0%

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

   5. *-un-lft-identity 52.0%

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

   6. fma-define 52.0%

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

   7. *-un-lft-identity 52.0%

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

   8. +-commutative 52.0%

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

   9. add-sqr-sqrt 39.7%

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

   10. sqrt-unprod 50.1%

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

   11. sqr-neg 50.1%

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

   12. sqrt-prod 10.6%

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

   13. add-sqr-sqrt 35.0%

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

   14. pow2 35.0%

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

   15. add-sqr-sqrt 24.6%

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

   16. sqrt-unprod 35.0%

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

   17. sqr-neg 35.0%

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

   18. sqrt-prod 10.6%

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

   19. add-sqr-sqrt 34.2%

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

   20. *-commutative 34.2%

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

   21. *-un-lft-identity 34.2%

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

6. Applied egg-rr 34.2%

   \[\leadsto \frac{\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(b, 1, b\right)}}{a \cdot 2} \]
7. Step-by-step derivation

   1. +-commutative 34.2%

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

   2. associate-+l+ 34.2%

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

   3. fma-undefine 34.2%

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

   4. *-rgt-identity 34.2%

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

8. Simplified 34.2%

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

9. Taylor expanded in b around inf 1.8%

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

11. Simplified 3.1%

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

12. Taylor expanded in b around 0 4.9%

    \[\leadsto \color{blue}{-0.5 \cdot c} \]
13. Step-by-step derivation

    1. *-commutative 4.9%

       \[\leadsto \color{blue}{c \cdot -0.5} \]

14. Simplified 4.9%

    \[\leadsto \color{blue}{c \cdot -0.5} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))