quadp (p42, positive)

Percentage Accurate: 52.3% → 82.6%

Time: 14.9s

Alternatives: 12

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(c \cdot a\right)\\ t_1 := t_0 + \mathsf{fma}\left(a \cdot -8, c, {b}^{2}\right)\\ \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{c \cdot -4} \cdot \sqrt{a}\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{t_1 - {b}^{2}}{b + \sqrt{t_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (* c a))) (t_1 (+ t_0 (fma (* a -8.0) c (pow b 2.0)))))
   (if (<= b -4.4e+55)
     (- (/ c b) (/ b a))
     (if (<= b 2.05e-235)
       (/ (- (sqrt (- (* b b) t_0)) b) (* a 2.0))
       (if (<= b 2.05e-90)
         (/ (- (hypot b (* (sqrt (* c -4.0)) (sqrt a))) b) (* a 2.0))
         (if (<= b 1e-40)
           (/ (/ (- t_1 (pow b 2.0)) (+ b (sqrt t_1))) (* a 2.0))
           (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * (c * a);
	double t_1 = t_0 + fma((a * -8.0), c, pow(b, 2.0));
	double tmp;
	if (b <= -4.4e+55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.05e-235) {
		tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	} else if (b <= 2.05e-90) {
		tmp = (hypot(b, (sqrt((c * -4.0)) * sqrt(a))) - b) / (a * 2.0);
	} else if (b <= 1e-40) {
		tmp = ((t_1 - pow(b, 2.0)) / (b + sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * Float64(c * a))
	t_1 = Float64(t_0 + fma(Float64(a * -8.0), c, (b ^ 2.0)))
	tmp = 0.0
	if (b <= -4.4e+55)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.05e-235)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 2.0));
	elseif (b <= 2.05e-90)
		tmp = Float64(Float64(hypot(b, Float64(sqrt(Float64(c * -4.0)) * sqrt(a))) - b) / Float64(a * 2.0));
	elseif (b <= 1e-40)
		tmp = Float64(Float64(Float64(t_1 - (b ^ 2.0)) / Float64(b + sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(a * -8.0), $MachinePrecision] * c + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.4e+55], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e-235], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e-90], N[(N[(N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(c * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-40], N[(N[(N[(t$95$1 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -4.40000000000000021e55

   1. Initial program 65.3%

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

   2. Taylor expanded in b around -inf 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

   4. Simplified 96.0%

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

   if -4.40000000000000021e55 < b < 2.04999999999999998e-235

   1. Initial program 86.9%

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

   if 2.04999999999999998e-235 < b < 2.05000000000000017e-90

   1. Initial program 39.1%

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

      1. +-commutative 39.1%

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

      2. add-sqr-sqrt 38.6%

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

      3. fma-def 38.7%

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

   3. Applied egg-rr 39.6%

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

      1. *-commutative 39.6%

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

      2. sqrt-prod 64.6%

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

   5. Applied egg-rr 64.6%

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

   if 2.05000000000000017e-90 < b < 9.9999999999999993e-41

   1. Initial program 68.9%

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

      1. remove-double-neg 68.9%

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

      2. distribute-frac-neg 68.9%

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

      3. distribute-neg-out 68.9%

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

      4. remove-double-neg 68.9%

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

      5. sub-neg 68.9%

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

      6. distribute-frac-neg 68.9%

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

      7. neg-mul-1 68.9%

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

   3. Simplified 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. associate--r- 68.9%

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

      2. count-2 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in b around 0 68.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]
   9. Step-by-step derivation

      1. flip-- 68.7%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} - b \cdot b}{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} + b}}}{a \cdot 2} \]

      2. add-sqr-sqrt 69.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)\right)} - b \cdot b}{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2} \]

      3. cancel-sign-sub-inv 69.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)\right)} - b \cdot b}{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2} \]

      4. associate-*r* 69.7%

         \[\leadsto \frac{\frac{\left(\left(\color{blue}{\left(-8 \cdot a\right) \cdot c} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)\right) - b \cdot b}{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2} \]

      5. fma-def 69.7%

         \[\leadsto \frac{\frac{\left(\color{blue}{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)\right) - b \cdot b}{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2} \]

      6. metadata-eval 69.7%

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)\right) - b \cdot b}{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2} \]

      7. unpow2 69.7%

         \[\leadsto \frac{\frac{\left(\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right) - \color{blue}{{b}^{2}}}{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} + b}}{a \cdot 2} \]

   10. Applied egg-rr 69.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b}}}{a \cdot 2} \]

   if 9.9999999999999993e-41 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   4. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{c \cdot -4} \cdot \sqrt{a}\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot \left(c \cdot a\right) + \mathsf{fma}\left(a \cdot -8, c, {b}^{2}\right)\right) - {b}^{2}}{b + \sqrt{4 \cdot \left(c \cdot a\right) + \mathsf{fma}\left(a \cdot -8, c, {b}^{2}\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 82.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{c \cdot -4} \cdot \sqrt{a}\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}} \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (pow b 2.0))))
   (if (<= b -3.1e+55)
     (- (/ c b) (/ b a))
     (if (<= b 2.05e-235)
       (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
       (if (<= b 2.05e-90)
         (/ (- (hypot b (* (sqrt (* c -4.0)) (sqrt a))) b) (* a 2.0))
         (if (<= b 1.4e-40)
           (* (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (/ 1.0 (* a 2.0)))
           (/ (- c) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), pow(b, 2.0));
	double tmp;
	if (b <= -3.1e+55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.05e-235) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else if (b <= 2.05e-90) {
		tmp = (hypot(b, (sqrt((c * -4.0)) * sqrt(a))) - b) / (a * 2.0);
	} else if (b <= 1.4e-40) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) * (1.0 / (a * 2.0));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), (b ^ 2.0))
	tmp = 0.0
	if (b <= -3.1e+55)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.05e-235)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	elseif (b <= 2.05e-90)
		tmp = Float64(Float64(hypot(b, Float64(sqrt(Float64(c * -4.0)) * sqrt(a))) - b) / Float64(a * 2.0));
	elseif (b <= 1.4e-40)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) * Float64(1.0 / Float64(a * 2.0)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.1e+55], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e-235], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e-90], N[(N[(N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(c * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-40], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -3.09999999999999994e55

   1. Initial program 65.3%

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

   2. Taylor expanded in b around -inf 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

   4. Simplified 96.0%

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

   if -3.09999999999999994e55 < b < 2.04999999999999998e-235

   1. Initial program 86.9%

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

   if 2.04999999999999998e-235 < b < 2.05000000000000017e-90

   1. Initial program 39.1%

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

      1. +-commutative 39.1%

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

      2. add-sqr-sqrt 38.6%

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

      3. fma-def 38.7%

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

   3. Applied egg-rr 39.6%

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

      1. *-commutative 39.6%

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

      2. sqrt-prod 64.6%

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

   5. Applied egg-rr 64.6%

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

   if 2.05000000000000017e-90 < b < 1.4e-40

   1. Initial program 68.9%

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

      1. remove-double-neg 68.9%

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

      2. distribute-frac-neg 68.9%

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

      3. distribute-neg-out 68.9%

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

      4. remove-double-neg 68.9%

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

      5. sub-neg 68.9%

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

      6. distribute-frac-neg 68.9%

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

      7. neg-mul-1 68.9%

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

   3. Simplified 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. associate--r- 68.9%

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

      2. count-2 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in b around 0 68.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]
   9. Step-by-step derivation

      1. div-inv 69.1%

         \[\leadsto \color{blue}{\left(\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2}} \]

      2. cancel-sign-sub-inv 69.1%

         \[\leadsto \left(\sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{1}{a \cdot 2} \]

      3. associate-*r* 69.1%

         \[\leadsto \left(\sqrt{\left(\color{blue}{\left(-8 \cdot a\right) \cdot c} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2} \]

      4. fma-def 69.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2} \]

      5. metadata-eval 69.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2} \]

   10. Applied egg-rr 69.1%

       \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2}} \]
   11. Step-by-step derivation

       1. flip-- 68.9%

          \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b}} \cdot \frac{1}{a \cdot 2} \]

       2. add-sqr-sqrt 69.6%

          \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       3. +-commutative 69.6%

          \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right) + \mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right)\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       4. fma-def 69.6%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right)\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       5. fma-udef 69.6%

          \[\leadsto \frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{\left(-8 \cdot a\right) \cdot c + {b}^{2}}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       6. associate-*r* 69.6%

          \[\leadsto \frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{-8 \cdot \left(a \cdot c\right)} + {b}^{2}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       7. fma-def 69.6%

          \[\leadsto \frac{\mathsf{fma}\left(4, a \cdot c, \color{blue}{\mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)}\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       8. unpow2 69.6%

          \[\leadsto \frac{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right) - \color{blue}{{b}^{2}}}{\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} + b} \cdot \frac{1}{a \cdot 2} \]

   12. Applied egg-rr 69.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right) - {b}^{2}}{\sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} + b}} \cdot \frac{1}{a \cdot 2} \]
   13. Step-by-step derivation

       1. fma-udef 69.6%

          \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right) + \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} - {b}^{2}}{\sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       2. fma-udef 69.6%

          \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right) + \color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right)}\right) - {b}^{2}}{\sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       3. associate-+r+ 69.6%

          \[\leadsto \frac{\color{blue}{\left(\left(4 \cdot \left(a \cdot c\right) + -8 \cdot \left(a \cdot c\right)\right) + {b}^{2}\right)} - {b}^{2}}{\sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       4. distribute-rgt-out 69.6%

          \[\leadsto \frac{\left(\color{blue}{\left(a \cdot c\right) \cdot \left(4 + -8\right)} + {b}^{2}\right) - {b}^{2}}{\sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       5. metadata-eval 69.6%

          \[\leadsto \frac{\left(\left(a \cdot c\right) \cdot \color{blue}{-4} + {b}^{2}\right) - {b}^{2}}{\sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       6. associate-*r* 69.6%

          \[\leadsto \frac{\left(\color{blue}{a \cdot \left(c \cdot -4\right)} + {b}^{2}\right) - {b}^{2}}{\sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       7. fma-def 69.6%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} - {b}^{2}}{\sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)} + b} \cdot \frac{1}{a \cdot 2} \]

       8. +-commutative 69.6%

          \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{\color{blue}{b + \sqrt{\mathsf{fma}\left(4, a \cdot c, \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)\right)}}} \cdot \frac{1}{a \cdot 2} \]

       9. fma-udef 69.6%

          \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\color{blue}{4 \cdot \left(a \cdot c\right) + \mathsf{fma}\left(-8, a \cdot c, {b}^{2}\right)}}} \cdot \frac{1}{a \cdot 2} \]

       10. fma-udef 69.6%

           \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{4 \cdot \left(a \cdot c\right) + \color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right)}}} \cdot \frac{1}{a \cdot 2} \]

       11. associate-+r+ 69.6%

           \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\color{blue}{\left(4 \cdot \left(a \cdot c\right) + -8 \cdot \left(a \cdot c\right)\right) + {b}^{2}}}} \cdot \frac{1}{a \cdot 2} \]

       12. distribute-rgt-out 69.6%

           \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(4 + -8\right)} + {b}^{2}}} \cdot \frac{1}{a \cdot 2} \]

       13. metadata-eval 69.6%

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

       14. associate-*r* 69.6%

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

       15. fma-def 69.6%

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

   14. Simplified 69.6%

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

   if 1.4e-40 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   4. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{c \cdot -4} \cdot \sqrt{a}\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}} \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{c \cdot -4} \cdot \sqrt{a}\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 10^{-40}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\sqrt{{b}^{2} + c \cdot \left(a \cdot -8 + a \cdot 4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e+55)
   (- (/ c b) (/ b a))
   (if (<= b 1.8e-235)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (if (<= b 2.05e-90)
       (/ (- (hypot b (* (sqrt (* c -4.0)) (sqrt a))) b) (* a 2.0))
       (if (<= b 1e-40)
         (*
          (/ 1.0 (* a 2.0))
          (- (sqrt (+ (pow b 2.0) (* c (+ (* a -8.0) (* a 4.0))))) b))
         (/ (- c) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e+55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.8e-235) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else if (b <= 2.05e-90) {
		tmp = (hypot(b, (sqrt((c * -4.0)) * sqrt(a))) - b) / (a * 2.0);
	} else if (b <= 1e-40) {
		tmp = (1.0 / (a * 2.0)) * (sqrt((pow(b, 2.0) + (c * ((a * -8.0) + (a * 4.0))))) - b);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e+55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.8e-235) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else if (b <= 2.05e-90) {
		tmp = (Math.hypot(b, (Math.sqrt((c * -4.0)) * Math.sqrt(a))) - b) / (a * 2.0);
	} else if (b <= 1e-40) {
		tmp = (1.0 / (a * 2.0)) * (Math.sqrt((Math.pow(b, 2.0) + (c * ((a * -8.0) + (a * 4.0))))) - b);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e+55:
		tmp = (c / b) - (b / a)
	elif b <= 1.8e-235:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	elif b <= 2.05e-90:
		tmp = (math.hypot(b, (math.sqrt((c * -4.0)) * math.sqrt(a))) - b) / (a * 2.0)
	elif b <= 1e-40:
		tmp = (1.0 / (a * 2.0)) * (math.sqrt((math.pow(b, 2.0) + (c * ((a * -8.0) + (a * 4.0))))) - b)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e+55)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.8e-235)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	elseif (b <= 2.05e-90)
		tmp = Float64(Float64(hypot(b, Float64(sqrt(Float64(c * -4.0)) * sqrt(a))) - b) / Float64(a * 2.0));
	elseif (b <= 1e-40)
		tmp = Float64(Float64(1.0 / Float64(a * 2.0)) * Float64(sqrt(Float64((b ^ 2.0) + Float64(c * Float64(Float64(a * -8.0) + Float64(a * 4.0))))) - b));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.4e+55)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.8e-235)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	elseif (b <= 2.05e-90)
		tmp = (hypot(b, (sqrt((c * -4.0)) * sqrt(a))) - b) / (a * 2.0);
	elseif (b <= 1e-40)
		tmp = (1.0 / (a * 2.0)) * (sqrt(((b ^ 2.0) + (c * ((a * -8.0) + (a * 4.0))))) - b);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e+55], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-235], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e-90], N[(N[(N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(c * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-40], N[(N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + N[(c * N[(N[(a * -8.0), $MachinePrecision] + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -4.40000000000000021e55

   1. Initial program 65.3%

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

   2. Taylor expanded in b around -inf 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

   4. Simplified 96.0%

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

   if -4.40000000000000021e55 < b < 1.79999999999999999e-235

   1. Initial program 86.9%

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

   if 1.79999999999999999e-235 < b < 2.05000000000000017e-90

   1. Initial program 39.1%

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

      1. +-commutative 39.1%

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

      2. add-sqr-sqrt 38.6%

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

      3. fma-def 38.7%

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

   3. Applied egg-rr 39.6%

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

      1. *-commutative 39.6%

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

      2. sqrt-prod 64.6%

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

   5. Applied egg-rr 64.6%

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

   if 2.05000000000000017e-90 < b < 9.9999999999999993e-41

   1. Initial program 68.9%

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

      1. remove-double-neg 68.9%

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

      2. distribute-frac-neg 68.9%

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

      3. distribute-neg-out 68.9%

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

      4. remove-double-neg 68.9%

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

      5. sub-neg 68.9%

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

      6. distribute-frac-neg 68.9%

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

      7. neg-mul-1 68.9%

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

   3. Simplified 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. associate--r- 68.9%

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

      2. count-2 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in b around 0 68.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]
   9. Step-by-step derivation

      1. div-inv 69.1%

         \[\leadsto \color{blue}{\left(\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2}} \]

      2. cancel-sign-sub-inv 69.1%

         \[\leadsto \left(\sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{1}{a \cdot 2} \]

      3. associate-*r* 69.1%

         \[\leadsto \left(\sqrt{\left(\color{blue}{\left(-8 \cdot a\right) \cdot c} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2} \]

      4. fma-def 69.1%

         \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2} \]

      5. metadata-eval 69.1%

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2} \]

   10. Applied egg-rr 69.1%

       \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-8 \cdot a, c, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2}} \]

   11. Taylor expanded in c around 0 69.1%

       \[\leadsto \left(\sqrt{\color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right) + {b}^{2}}} - b\right) \cdot \frac{1}{a \cdot 2} \]

   if 9.9999999999999993e-41 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   4. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{c \cdot -4} \cdot \sqrt{a}\right) - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 10^{-40}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\sqrt{{b}^{2} + c \cdot \left(a \cdot -8 + a \cdot 4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e+55)
   (- (/ c b) (/ b a))
   (if (<= b 1.1e-40)
     (/ (- (sqrt (fma 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 <= -4.4e+55) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-40) {
		tmp = (sqrt(fma(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 <= -4.4e+55)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.1e-40)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e+55], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-40], N[(N[(N[Sqrt[N[(b * b + 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 < -4.40000000000000021e55

   1. Initial program 65.3%

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

   2. Taylor expanded in b around -inf 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

   4. Simplified 96.0%

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

   if -4.40000000000000021e55 < b < 1.10000000000000004e-40

   1. Initial program 77.8%

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

      1. remove-double-neg 77.8%

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

      2. distribute-frac-neg 77.8%

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

      3. distribute-neg-out 77.8%

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

      4. remove-double-neg 77.8%

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

      5. sub-neg 77.8%

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

      6. distribute-frac-neg 77.8%

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

      7. neg-mul-1 77.8%

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

   3. Simplified 77.9%

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

   if 1.10000000000000004e-40 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+54)
   (- (/ c b) (/ b a))
   (if (<= b 9.5e-41)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9.5e-41) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - 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 <= (-2.2d+54)) then
        tmp = (c / b) - (b / a)
    else if (b <= 9.5d-41) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - 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 <= -2.2e+54) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9.5e-41) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e+54:
		tmp = (c / b) - (b / a)
	elif b <= 9.5e-41:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+54)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 9.5e-41)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e+54)
		tmp = (c / b) - (b / a);
	elseif (b <= 9.5e-41)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+54], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.5e-41], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $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 < -2.1999999999999999e54

   1. Initial program 65.3%

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

   2. Taylor expanded in b around -inf 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

   4. Simplified 96.0%

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

   if -2.1999999999999999e54 < b < 9.4999999999999997e-41

   1. Initial program 77.8%

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

   if 9.4999999999999997e-41 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-69}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-69)
   (/ (- b) a)
   (if (<= b 1.15e-40)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-69) {
		tmp = -b / a;
	} else if (b <= 1.15e-40) {
		tmp = (sqrt((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 <= (-8d-69)) then
        tmp = -b / a
    else if (b <= 1.15d-40) then
        tmp = (sqrt((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 <= -8e-69) {
		tmp = -b / a;
	} else if (b <= 1.15e-40) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8e-69:
		tmp = -b / a
	elif b <= 1.15e-40:
		tmp = (math.sqrt((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 <= -8e-69)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.15e-40)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8e-69)
		tmp = -b / a;
	elseif (b <= 1.15e-40)
		tmp = (sqrt((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, -8e-69], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.15e-40], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $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 < -7.9999999999999997e-69

   1. Initial program 70.8%

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

   2. Taylor expanded in b around -inf 88.8%

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

      1. associate-*r/ 88.8%

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

      2. mul-1-neg 88.8%

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

   4. Simplified 88.8%

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

   if -7.9999999999999997e-69 < b < 1.15e-40

   1. Initial program 75.5%

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

      1. remove-double-neg 75.5%

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

      2. distribute-frac-neg 75.5%

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

      3. distribute-neg-out 75.5%

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

      4. remove-double-neg 75.5%

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

      5. sub-neg 75.5%

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

      6. distribute-frac-neg 75.5%

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

      7. neg-mul-1 75.5%

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

   3. Simplified 75.5%

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

   5. Applied egg-rr 75.1%

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

      1. associate--r- 75.1%

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

      2. count-2 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in b around 0 65.9%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]
   9. Step-by-step derivation

      1. distribute-rgt-out-- 66.3%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 - -4\right)}} - b}{a \cdot 2} \]

      2. metadata-eval 66.3%

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

      3. *-commutative 66.3%

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

      4. associate-*r* 66.4%

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

      5. *-commutative 66.4%

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

      6. metadata-eval 66.4%

         \[\leadsto \frac{\sqrt{\left(a \cdot \color{blue}{\left(-8 - -4\right)}\right) \cdot c} - b}{a \cdot 2} \]

      7. distribute-rgt-out-- 66.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot a - -4 \cdot a\right)} \cdot c} - b}{a \cdot 2} \]

      8. *-commutative 66.4%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-8 \cdot a - -4 \cdot a\right)}} - b}{a \cdot 2} \]

      9. distribute-rgt-out-- 66.4%

         \[\leadsto \frac{\sqrt{c \cdot \color{blue}{\left(a \cdot \left(-8 - -4\right)\right)}} - b}{a \cdot 2} \]

      10. metadata-eval 66.4%

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

   10. Simplified 66.4%

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

   if 1.15e-40 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-69}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-119}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-119)
   (/ (- b) a)
   (if (<= b 1.35e-40) (* 0.5 (/ (sqrt (* a (* c -4.0))) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-119) {
		tmp = -b / a;
	} else if (b <= 1.35e-40) {
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / 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 <= (-1.1d-119)) then
        tmp = -b / a
    else if (b <= 1.35d-40) then
        tmp = 0.5d0 * (sqrt((a * (c * (-4.0d0)))) / 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 <= -1.1e-119) {
		tmp = -b / a;
	} else if (b <= 1.35e-40) {
		tmp = 0.5 * (Math.sqrt((a * (c * -4.0))) / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.1e-119:
		tmp = -b / a
	elif b <= 1.35e-40:
		tmp = 0.5 * (math.sqrt((a * (c * -4.0))) / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-119)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.35e-40)
		tmp = Float64(0.5 * Float64(sqrt(Float64(a * Float64(c * -4.0))) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e-119)
		tmp = -b / a;
	elseif (b <= 1.35e-40)
		tmp = 0.5 * (sqrt((a * (c * -4.0))) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-119], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.35e-40], N[(0.5 * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1e-119

   1. Initial program 72.8%

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

   2. Taylor expanded in b around -inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. mul-1-neg 84.8%

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

   4. Simplified 84.8%

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

   if -1.1e-119 < b < 1.35e-40

   1. Initial program 72.8%

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

      1. remove-double-neg 72.8%

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

      2. distribute-frac-neg 72.8%

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

      3. distribute-neg-out 72.8%

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

      4. remove-double-neg 72.8%

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

      5. sub-neg 72.8%

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

      6. distribute-frac-neg 72.8%

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

      7. neg-mul-1 72.8%

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

   3. Simplified 72.8%

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

   5. Applied egg-rr 72.4%

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

      1. associate--r- 72.4%

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

      2. count-2 72.4%

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

   7. Simplified 72.4%

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

   8. Taylor expanded in b around 0 72.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}} - b}{a \cdot 2} \]

   9. Taylor expanded in b around 0 67.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}\right)} \]
   10. Step-by-step derivation

       1. associate-*l/ 67.5%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot \sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}}{a}} \]

       2. *-lft-identity 67.5%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}}}{a} \]

       3. distribute-rgt-out-- 68.0%

          \[\leadsto 0.5 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-8 - -4\right)}}}{a} \]

       4. metadata-eval 68.0%

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

       5. associate-*r* 68.0%

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

   11. Simplified 68.0%

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

   if 1.35e-40 < b

   1. Initial program 15.6%

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

   2. Taylor expanded in b around inf 90.0%

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

      1. associate-*r/ 90.0%

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

      2. neg-mul-1 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-119}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 67.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-309) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-309:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-309)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-309)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-309], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 75.1%

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

   2. Taylor expanded in b around -inf 71.0%

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

      1. +-commutative 71.0%

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

      2. mul-1-neg 71.0%

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

      3. unsub-neg 71.0%

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

   4. Simplified 71.0%

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

   if -1.000000000000002e-309 < b

   1. Initial program 29.6%

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

   2. Taylor expanded in b around inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. neg-mul-1 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 9: 43.3% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 8.8e+46) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.8e+46) {
		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 <= 8.8d+46) 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 <= 8.8e+46) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.8e+46:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.8e+46)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.8e+46)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.8e+46], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.8000000000000001e46

   1. Initial program 67.5%

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

   2. Taylor expanded in b around -inf 52.8%

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

      1. associate-*r/ 52.8%

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

      2. mul-1-neg 52.8%

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

   4. Simplified 52.8%

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

   if 8.8000000000000001e46 < b

   1. Initial program 15.5%

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

   3. Applied egg-rr 4.1%

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

   4. Taylor expanded in b around -inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 29.4%

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

      4. associate-*r/ 29.4%

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

      5. *-rgt-identity 29.4%

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

      6. *-commutative 29.4%

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

      7. times-frac 29.6%

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

      8. /-rgt-identity 29.6%

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

      9. *-commutative 29.6%

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

      10. *-lft-identity 29.6%

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

      11. times-frac 29.6%

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

      12. metadata-eval 29.6%

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

   6. Simplified 29.6%

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

   7. Taylor expanded in a around 0 29.4%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 10: 67.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.6e-278) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.6e-278) {
		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 <= 3.6d-278) 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 <= 3.6e-278) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.6e-278:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.6e-278)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.6e-278)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.6e-278], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.59999999999999996e-278

   1. Initial program 75.3%

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

   2. Taylor expanded in b around -inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. mul-1-neg 70.5%

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

   4. Simplified 70.5%

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

   if 3.59999999999999996e-278 < b

   1. Initial program 29.0%

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

   2. Taylor expanded in b around inf 70.7%

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

      1. associate-*r/ 70.7%

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

      2. neg-mul-1 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 11: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

3. Applied egg-rr 22.3%

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

4. Taylor expanded in a around 0 2.4%

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

5. Final simplification 2.4%

   \[\leadsto \frac{b}{a} \]

Alternative 12: 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 54.7%

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

3. Applied egg-rr 22.3%

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

4. Taylor expanded in b around -inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 9.4%

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

   4. associate-*r/ 9.4%

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

   5. *-rgt-identity 9.4%

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

   6. *-commutative 9.4%

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

   7. times-frac 9.4%

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

   8. /-rgt-identity 9.4%

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

   9. *-commutative 9.4%

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

   10. *-lft-identity 9.4%

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

   11. times-frac 9.4%

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

   12. metadata-eval 9.4%

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

6. Simplified 9.4%

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

7. Taylor expanded in a around 0 9.4%

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

8. Final simplification 9.4%

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

Developer target: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

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

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