quadp (p42, positive)

Percentage Accurate: 52.9% → 83.9%

Time: 18.4s

Alternatives: 12

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% 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: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{0.5 \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot -4} \cdot \sqrt{c}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+159)
   (- (/ b a))
   (if (<= b 1.45e-102)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (if (or (<= b 1.25e-38) (not (<= b 5100000.0)))
       (/ (- c) b)
       (/ 1.0 (/ a (* 0.5 (+ b (hypot b (* (sqrt (* a -4.0)) (sqrt c)))))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+159) {
		tmp = -(b / a);
	} else if (b <= 1.45e-102) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if ((b <= 1.25e-38) || !(b <= 5100000.0)) {
		tmp = -c / b;
	} else {
		tmp = 1.0 / (a / (0.5 * (b + hypot(b, (sqrt((a * -4.0)) * sqrt(c))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+159)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.45e-102)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	elseif ((b <= 1.25e-38) || !(b <= 5100000.0))
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(1.0 / Float64(a / Float64(0.5 * Float64(b + hypot(b, Float64(sqrt(Float64(a * -4.0)) * sqrt(c)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+159], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.45e-102], 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], If[Or[LessEqual[b, 1.25e-38], N[Not[LessEqual[b, 5100000.0]], $MachinePrecision]], N[((-c) / b), $MachinePrecision], N[(1.0 / N[(a / N[(0.5 * N[(b + N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.2e159

   1. Initial program 56.7%

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

      1. *-commutative 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.2e159 < b < 1.44999999999999993e-102

   1. Initial program 83.4%

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

      1. +-commutative 83.4%

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

      2. unsub-neg 83.4%

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

      3. fma-neg 83.4%

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

      4. distribute-lft-neg-in 83.4%

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

      5. *-commutative 83.4%

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

      6. *-commutative 83.4%

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

      7. associate-*l* 83.4%

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

      8. metadata-eval 83.4%

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

      9. *-commutative 83.4%

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

   3. Simplified 83.4%

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

   if 1.44999999999999993e-102 < b < 1.25000000000000008e-38 or 5.1e6 < b

   1. Initial program 13.7%

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

      1. *-commutative 13.7%

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

   3. Simplified 13.7%

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

   5. Taylor expanded in b around inf 89.0%

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

      1. associate-*r/ 89.0%

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

      2. neg-mul-1 89.0%

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

   7. Simplified 89.0%

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

   if 1.25000000000000008e-38 < b < 5.1e6

   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%

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

   3. Simplified 39.1%

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

   6. Applied egg-rr 39.3%

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

      1. sub-neg 39.3%

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

      2. distribute-rgt-out-- 39.3%

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

      3. *-commutative 39.3%

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

   8. Simplified 39.3%

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

      1. associate-*l/ 39.1%

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

      2. clear-num 39.1%

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

      3. sub-neg 39.1%

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

      4. sqrt-prod 12.5%

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

      5. *-commutative 12.5%

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

      6. sqrt-prod 39.1%

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

      7. associate-*r* 39.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 39.2%

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

      10. sqr-neg 39.2%

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

      11. sqrt-unprod 39.2%

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

      12. add-sqr-sqrt 39.2%

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

   10. Applied egg-rr 39.2%

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

       1. sqrt-prod 75.3%

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

   12. Applied egg-rr 75.3%

       \[\leadsto \frac{1}{\frac{a}{0.5 \cdot \left(\mathsf{hypot}\left(b, \color{blue}{\sqrt{a \cdot -4} \cdot \sqrt{c}}\right) + b\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{0.5 \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot -4} \cdot \sqrt{c}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot -4} \cdot \sqrt{c}\right) - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+159)
   (- (/ b a))
   (if (<= b 1.95e-103)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (if (or (<= b 1.25e-38) (not (<= b 5100000.0)))
       (/ (- c) b)
       (* (/ 0.5 a) (- (hypot b (* (sqrt (* a -4.0)) (sqrt c))) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+159) {
		tmp = -(b / a);
	} else if (b <= 1.95e-103) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if ((b <= 1.25e-38) || !(b <= 5100000.0)) {
		tmp = -c / b;
	} else {
		tmp = (0.5 / a) * (hypot(b, (sqrt((a * -4.0)) * sqrt(c))) - b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+159)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.95e-103)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	elseif ((b <= 1.25e-38) || !(b <= 5100000.0))
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(0.5 / a) * Float64(hypot(b, Float64(sqrt(Float64(a * -4.0)) * sqrt(c))) - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+159], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.95e-103], 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], If[Or[LessEqual[b, 1.25e-38], N[Not[LessEqual[b, 5100000.0]], $MachinePrecision]], N[((-c) / b), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.2e159

   1. Initial program 56.7%

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

      1. *-commutative 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.2e159 < b < 1.9500000000000001e-103

   1. Initial program 83.4%

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

      1. +-commutative 83.4%

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

      2. unsub-neg 83.4%

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

      3. fma-neg 83.4%

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

      4. distribute-lft-neg-in 83.4%

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

      5. *-commutative 83.4%

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

      6. *-commutative 83.4%

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

      7. associate-*l* 83.4%

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

      8. metadata-eval 83.4%

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

      9. *-commutative 83.4%

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

   3. Simplified 83.4%

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

   if 1.9500000000000001e-103 < b < 1.25000000000000008e-38 or 5.1e6 < b

   1. Initial program 13.7%

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

      1. *-commutative 13.7%

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

   3. Simplified 13.7%

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

   5. Taylor expanded in b around inf 89.0%

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

      1. associate-*r/ 89.0%

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

      2. neg-mul-1 89.0%

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

   7. Simplified 89.0%

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

   if 1.25000000000000008e-38 < b < 5.1e6

   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%

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

   3. Simplified 39.1%

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

   6. Applied egg-rr 39.3%

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

      1. sub-neg 39.3%

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

      2. distribute-rgt-out-- 39.3%

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

      3. *-commutative 39.3%

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

   8. Simplified 39.3%

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

      1. sqrt-prod 12.5%

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

      2. *-commutative 12.5%

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

      3. sqrt-prod 38.8%

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

      4. pow1/2 38.8%

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

      5. associate-*r* 38.8%

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

      6. unpow-prod-down 75.0%

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

      7. pow1/2 75.0%

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

   10. Applied egg-rr 75.3%

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

       1. unpow1/2 75.0%

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

   12. Simplified 75.3%

       \[\leadsto \frac{0.5}{a} \cdot \left(\mathsf{hypot}\left(b, \color{blue}{\sqrt{a \cdot -4} \cdot \sqrt{c}}\right) - b\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\mathsf{hypot}\left(b, \sqrt{a \cdot -4} \cdot \sqrt{c}\right) - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \sqrt{c} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+159)
   (- (/ b a))
   (if (<= b 3.4e-102)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (or (<= b 1.25e-38) (not (<= b 5100000.0)))
       (/ (- c) b)
       (/ (- (* (sqrt (* a -4.0)) (sqrt c)) b) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+159) {
		tmp = -(b / a);
	} else if (b <= 3.4e-102) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if ((b <= 1.25e-38) || !(b <= 5100000.0)) {
		tmp = -c / b;
	} else {
		tmp = ((sqrt((a * -4.0)) * sqrt(c)) - b) / (a * 2.0);
	}
	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 <= (-1.2d+159)) then
        tmp = -(b / a)
    else if (b <= 3.4d-102) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else if ((b <= 1.25d-38) .or. (.not. (b <= 5100000.0d0))) then
        tmp = -c / b
    else
        tmp = ((sqrt((a * (-4.0d0))) * sqrt(c)) - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+159) {
		tmp = -(b / a);
	} else if (b <= 3.4e-102) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if ((b <= 1.25e-38) || !(b <= 5100000.0)) {
		tmp = -c / b;
	} else {
		tmp = ((Math.sqrt((a * -4.0)) * Math.sqrt(c)) - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.2e+159:
		tmp = -(b / a)
	elif b <= 3.4e-102:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	elif (b <= 1.25e-38) or not (b <= 5100000.0):
		tmp = -c / b
	else:
		tmp = ((math.sqrt((a * -4.0)) * math.sqrt(c)) - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+159)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 3.4e-102)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif ((b <= 1.25e-38) || !(b <= 5100000.0))
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(a * -4.0)) * sqrt(c)) - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.2e+159)
		tmp = -(b / a);
	elseif (b <= 3.4e-102)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	elseif ((b <= 1.25e-38) || ~((b <= 5100000.0)))
		tmp = -c / b;
	else
		tmp = ((sqrt((a * -4.0)) * sqrt(c)) - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+159], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 3.4e-102], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1.25e-38], N[Not[LessEqual[b, 5100000.0]], $MachinePrecision]], N[((-c) / b), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.2e159

   1. Initial program 56.7%

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

      1. *-commutative 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.2e159 < b < 3.40000000000000013e-102

   1. Initial program 83.4%

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

   if 3.40000000000000013e-102 < b < 1.25000000000000008e-38 or 5.1e6 < b

   1. Initial program 13.7%

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

      1. *-commutative 13.7%

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

   3. Simplified 13.7%

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

   5. Taylor expanded in b around inf 89.0%

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

      1. associate-*r/ 89.0%

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

      2. neg-mul-1 89.0%

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

   7. Simplified 89.0%

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

   if 1.25000000000000008e-38 < b < 5.1e6

   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%

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

   3. Simplified 39.1%

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

   5. Taylor expanded in b around 0 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-*r* 38.8%

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

   7. Simplified 38.8%

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

      1. sqrt-prod 12.5%

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

      2. *-commutative 12.5%

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

      3. sqrt-prod 38.8%

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

      4. pow1/2 38.8%

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

      5. associate-*r* 38.8%

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

      6. unpow-prod-down 75.0%

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

      7. pow1/2 75.0%

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

   9. Applied egg-rr 75.0%

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

       1. unpow1/2 75.0%

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

   11. Simplified 75.0%

       \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{a \cdot -4} \cdot \sqrt{c}}}{a \cdot 2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \sqrt{c} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \sqrt{c} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+159)
   (- (/ b a))
   (if (<= b 2.5e-102)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (if (or (<= b 1.25e-38) (not (<= b 5100000.0)))
       (/ (- c) b)
       (/ (- (* (sqrt (* a -4.0)) (sqrt c)) b) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+159) {
		tmp = -(b / a);
	} else if (b <= 2.5e-102) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else if ((b <= 1.25e-38) || !(b <= 5100000.0)) {
		tmp = -c / b;
	} else {
		tmp = ((sqrt((a * -4.0)) * sqrt(c)) - b) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+159)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 2.5e-102)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	elseif ((b <= 1.25e-38) || !(b <= 5100000.0))
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(a * -4.0)) * sqrt(c)) - b) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+159], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 2.5e-102], 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], If[Or[LessEqual[b, 1.25e-38], N[Not[LessEqual[b, 5100000.0]], $MachinePrecision]], N[((-c) / b), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.2e159

   1. Initial program 56.7%

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

      1. *-commutative 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.2e159 < b < 2.50000000000000013e-102

   1. Initial program 83.4%

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

      1. +-commutative 83.4%

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

      2. unsub-neg 83.4%

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

      3. fma-neg 83.4%

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

      4. distribute-lft-neg-in 83.4%

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

      5. *-commutative 83.4%

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

      6. *-commutative 83.4%

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

      7. associate-*l* 83.4%

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

      8. metadata-eval 83.4%

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

      9. *-commutative 83.4%

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

   3. Simplified 83.4%

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

   if 2.50000000000000013e-102 < b < 1.25000000000000008e-38 or 5.1e6 < b

   1. Initial program 13.7%

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

      1. *-commutative 13.7%

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

   3. Simplified 13.7%

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

   5. Taylor expanded in b around inf 89.0%

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

      1. associate-*r/ 89.0%

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

      2. neg-mul-1 89.0%

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

   7. Simplified 89.0%

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

   if 1.25000000000000008e-38 < b < 5.1e6

   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%

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

   3. Simplified 39.1%

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

   5. Taylor expanded in b around 0 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-*r* 38.8%

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

   7. Simplified 38.8%

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

      1. sqrt-prod 12.5%

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

      2. *-commutative 12.5%

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

      3. sqrt-prod 38.8%

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

      4. pow1/2 38.8%

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

      5. associate-*r* 38.8%

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

      6. unpow-prod-down 75.0%

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

      7. pow1/2 75.0%

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

   9. Applied egg-rr 75.0%

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

       1. unpow1/2 75.0%

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

   11. Simplified 75.0%

       \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{a \cdot -4} \cdot \sqrt{c}}}{a \cdot 2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-38} \lor \neg \left(b \leq 5100000\right):\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \sqrt{c} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+159)
   (- (/ b a))
   (if (<= b 4.8e-104)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.2e+159) {
		tmp = -(b / a);
	} else if (b <= 4.8e-104) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - 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 <= (-1.2d+159)) then
        tmp = -(b / a)
    else if (b <= 4.8d-104) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - 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 <= -1.2e+159) {
		tmp = -(b / a);
	} else if (b <= 4.8e-104) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.2e+159:
		tmp = -(b / a)
	elif b <= 4.8e-104:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.2e+159)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 4.8e-104)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - 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 <= -1.2e+159)
		tmp = -(b / a);
	elseif (b <= 4.8e-104)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+159], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 4.8e-104], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $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 < -1.2e159

   1. Initial program 56.7%

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

      1. *-commutative 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1.2e159 < b < 4.8000000000000001e-104

   1. Initial program 83.4%

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

   if 4.8000000000000001e-104 < 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. Step-by-step derivation

      1. *-commutative 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. neg-mul-1 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+159}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-103}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e-48)
   (- (/ c b) (/ b a))
   (if (<= b 1.95e-103)
     (* (/ 0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.8e-48) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-103) {
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} 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 <= (-5.8d-48)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.95d-103) then
        tmp = (0.5d0 / a) * (b + sqrt((c * (a * (-4.0d0)))))
    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 <= -5.8e-48) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.95e-103) {
		tmp = (0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.8e-48:
		tmp = (c / b) - (b / a)
	elif b <= 1.95e-103:
		tmp = (0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.8e-48)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.95e-103)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.8e-48)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.95e-103)
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.8e-48], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-103], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.8000000000000006e-48

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in b around -inf 83.8%

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

      1. +-commutative 83.8%

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

      2. mul-1-neg 83.8%

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

      3. unsub-neg 83.8%

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

   7. Simplified 83.8%

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

   if -5.8000000000000006e-48 < b < 1.9500000000000001e-103

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in b around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. associate-*r* 70.6%

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

   7. Simplified 70.6%

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

      1. expm1-log1p-u 45.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)\right)} \]

      2. expm1-udef 17.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)} - 1} \]

      3. *-un-lft-identity 17.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}{a \cdot 2}\right)} - 1 \]

      4. *-commutative 17.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \left(\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{\color{blue}{2 \cdot a}}\right)} - 1 \]

      5. times-frac 17.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}}\right)} - 1 \]

      6. metadata-eval 17.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5} \cdot \frac{\left(-b\right) + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\right)} - 1 \]

      7. add-sqr-sqrt 12.2%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\right)} - 1 \]

      8. sqrt-unprod 17.0%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\right)} - 1 \]

      9. sqr-neg 17.0%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\right)} - 1 \]

      10. sqrt-unprod 5.1%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\right)} - 1 \]

      11. add-sqr-sqrt 16.3%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \frac{\color{blue}{b} + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\right)} - 1 \]

   9. Applied egg-rr 16.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 44.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\right)\right)} \]

       2. expm1-log1p 69.2%

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

       3. *-commutative 69.2%

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

       4. associate-*l/ 69.2%

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

       5. associate-*r/ 69.0%

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

       6. *-commutative 69.0%

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

       7. associate-*r* 69.0%

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

       8. *-commutative 69.0%

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

       9. associate-*l* 69.0%

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

   11. Simplified 69.0%

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

   if 1.9500000000000001e-103 < 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. Step-by-step derivation

      1. *-commutative 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. neg-mul-1 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-103}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.8e-48)
   (- (/ c b) (/ b a))
   (if (<= b 2e-103) (/ (- (sqrt (* 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-48) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2e-103) {
		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 <= (-8.8d-48)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2d-103) 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 <= -8.8e-48) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2e-103) {
		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 <= -8.8e-48:
		tmp = (c / b) - (b / a)
	elif b <= 2e-103:
		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 <= -8.8e-48)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2e-103)
		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 <= -8.8e-48)
		tmp = (c / b) - (b / a);
	elseif (b <= 2e-103)
		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, -8.8e-48], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-103], 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 < -8.8000000000000005e-48

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in b around -inf 83.8%

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

      1. +-commutative 83.8%

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

      2. mul-1-neg 83.8%

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

      3. unsub-neg 83.8%

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

   7. Simplified 83.8%

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

   if -8.8000000000000005e-48 < b < 1.99999999999999992e-103

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in b around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. associate-*r* 70.6%

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

   7. Simplified 70.6%

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

      1. +-commutative 70.6%

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

      2. unsub-neg 70.6%

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

   9. Applied egg-rr 70.6%

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

       1. associate-*r* 70.6%

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

       2. *-commutative 70.6%

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

       3. associate-*l* 70.6%

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

   11. Simplified 70.6%

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

   if 1.99999999999999992e-103 < 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. Step-by-step derivation

      1. *-commutative 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in b around inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. neg-mul-1 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		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 <= -5e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 81.5%

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

      1. *-commutative 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in b around -inf 63.8%

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

      1. +-commutative 63.8%

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

      2. mul-1-neg 63.8%

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

      3. unsub-neg 63.8%

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

   7. Simplified 63.8%

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

   if -4.999999999999985e-310 < b

   1. Initial program 27.1%

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

      1. *-commutative 27.1%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in b around inf 66.0%

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

      1. associate-*r/ 66.0%

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

      2. neg-mul-1 66.0%

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

   7. Simplified 66.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.6e+21) (- (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6e+21) {
		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 <= 1.6d+21) 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 <= 1.6e+21) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.6e+21:
		tmp = -(b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6e+21)
		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 <= 1.6e+21)
		tmp = -(b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6e+21], (-N[(b / a), $MachinePrecision]), N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6e21

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in b around -inf 42.7%

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

      1. associate-*r/ 42.7%

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

      2. mul-1-neg 42.7%

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

   7. Simplified 42.7%

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

   if 1.6e21 < b

   1. Initial program 6.4%

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

      1. *-commutative 6.4%

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

   3. Simplified 6.4%

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

   6. Applied egg-rr 8.8%

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

      1. sub-neg 8.8%

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

      2. distribute-rgt-out-- 14.8%

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

      3. *-commutative 14.8%

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

   8. Simplified 14.8%

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

   9. Taylor expanded in b around -inf 0.0%

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

       1. +-commutative 0.0%

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

       2. mul-1-neg 0.0%

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

       3. unsub-neg 0.0%

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

       4. *-commutative 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 2.3%

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

       7. *-commutative 2.3%

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

   11. Simplified 2.3%

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

   12. Taylor expanded in a around inf 23.9%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2e-281) (- (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2e-281) {
		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 <= 5.2d-281) 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 <= 5.2e-281) {
		tmp = -(b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.2e-281:
		tmp = -(b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2e-281)
		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 <= 5.2e-281)
		tmp = -(b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2e-281], (-N[(b / a), $MachinePrecision]), N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2000000000000001e-281

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in b around -inf 62.4%

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

      1. associate-*r/ 62.4%

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

      2. mul-1-neg 62.4%

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

   7. Simplified 62.4%

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

   if 5.2000000000000001e-281 < b

   1. Initial program 27.5%

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

      1. *-commutative 27.5%

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

   3. Simplified 27.5%

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

   5. Taylor expanded in b around inf 66.9%

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

      1. associate-*r/ 66.9%

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

      2. neg-mul-1 66.9%

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

   7. Simplified 66.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-281}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 2.6% 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 51.5%

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

   1. *-commutative 51.5%

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

3. Simplified 51.5%

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

6. Applied egg-rr 46.2%

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

   1. sub-neg 46.2%

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

   2. distribute-rgt-out-- 48.0%

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

   3. *-commutative 48.0%

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

8. Simplified 48.0%

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

   1. associate-*l/ 48.1%

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

   2. clear-num 48.0%

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

   3. sub-neg 48.0%

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

   4. sqrt-prod 27.0%

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

   5. *-commutative 27.0%

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

   6. sqrt-prod 48.0%

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

   7. associate-*r* 48.0%

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

   8. add-sqr-sqrt 30.4%

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

   9. sqrt-unprod 41.9%

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

   10. sqr-neg 41.9%

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

   11. sqrt-unprod 12.9%

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

   12. add-sqr-sqrt 26.9%

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

10. Applied egg-rr 26.9%

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

11. Taylor expanded in a around 0 2.6%

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

12. Final simplification 2.6%

    \[\leadsto \frac{b}{a} \]
13. Add Preprocessing

Alternative 12: 11.0% 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 51.5%

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

   1. *-commutative 51.5%

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

3. Simplified 51.5%

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

6. Applied egg-rr 46.2%

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

   1. sub-neg 46.2%

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

   2. distribute-rgt-out-- 48.0%

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

   3. *-commutative 48.0%

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

8. Simplified 48.0%

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

9. Taylor expanded in b around -inf 0.0%

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

    1. +-commutative 0.0%

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

    2. mul-1-neg 0.0%

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

    3. unsub-neg 0.0%

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

    4. *-commutative 0.0%

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

    5. unpow2 0.0%

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

    6. rem-square-sqrt 28.9%

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

    7. *-commutative 28.9%

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

11. Simplified 28.9%

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

12. Taylor expanded in a around inf 9.8%

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

13. Final simplification 9.8%

    \[\leadsto \frac{c}{b} \]
14. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

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

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 2024027 
(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)))