The quadratic formula (r1)

Percentage Accurate: 53.8% → 85.1%

Time: 19.3s

Alternatives: 11

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot -4} \cdot \sqrt{a}\right)\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (- b) a)
   (if (<= b -3.8e-20)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b -2.55e-48)
       (* (- b (hypot b (* (sqrt (* c -4.0)) (sqrt a)))) (/ -0.5 a))
       (if (<= b 5.2e-70)
         (* (/ -0.5 a) (- b (hypot b (sqrt (* a (* c -4.0))))))
         (/ (- c) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+153) {
		tmp = -b / a;
	} else if (b <= -3.8e-20) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= -2.55e-48) {
		tmp = (b - hypot(b, (sqrt((c * -4.0)) * sqrt(a)))) * (-0.5 / a);
	} else if (b <= 5.2e-70) {
		tmp = (-0.5 / a) * (b - hypot(b, sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+153) {
		tmp = -b / a;
	} else if (b <= -3.8e-20) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= -2.55e-48) {
		tmp = (b - Math.hypot(b, (Math.sqrt((c * -4.0)) * Math.sqrt(a)))) * (-0.5 / a);
	} else if (b <= 5.2e-70) {
		tmp = (-0.5 / a) * (b - Math.hypot(b, Math.sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+153:
		tmp = -b / a
	elif b <= -3.8e-20:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	elif b <= -2.55e-48:
		tmp = (b - math.hypot(b, (math.sqrt((c * -4.0)) * math.sqrt(a)))) * (-0.5 / a)
	elif b <= 5.2e-70:
		tmp = (-0.5 / a) * (b - math.hypot(b, math.sqrt((a * (c * -4.0)))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+153)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -3.8e-20)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= -2.55e-48)
		tmp = Float64(Float64(b - hypot(b, Float64(sqrt(Float64(c * -4.0)) * sqrt(a)))) * Float64(-0.5 / a));
	elseif (b <= 5.2e-70)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - hypot(b, sqrt(Float64(a * Float64(c * -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 <= -1e+153)
		tmp = -b / a;
	elseif (b <= -3.8e-20)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	elseif (b <= -2.55e-48)
		tmp = (b - hypot(b, (sqrt((c * -4.0)) * sqrt(a)))) * (-0.5 / a);
	elseif (b <= 5.2e-70)
		tmp = (-0.5 / a) * (b - hypot(b, sqrt((a * (c * -4.0)))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+153], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -3.8e-20], 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[LessEqual[b, -2.55e-48], N[(N[(b - N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(c * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-70], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1e153

   1. Initial program 38.1%

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

      1. sqr-neg 38.1%

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

      2. sqr-neg 38.1%

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

      3. associate-*l* 38.1%

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

      4. *-commutative 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in b around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. mul-1-neg 97.7%

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

   6. Simplified 97.7%

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

   if -1e153 < b < -3.7999999999999998e-20

   1. Initial program 97.1%

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

      1. sqr-neg 97.1%

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

      2. sqr-neg 97.1%

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

      3. associate-*l* 97.1%

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

      4. *-commutative 97.1%

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

   3. Simplified 97.1%

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

   if -3.7999999999999998e-20 < b < -2.55000000000000006e-48

   1. Initial program 35.6%

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

      1. sqr-neg 35.6%

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

      2. sqr-neg 35.6%

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

      3. associate-*l* 35.6%

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

      4. *-commutative 35.6%

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

   3. Simplified 35.6%

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

      1. frac-2neg 35.6%

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

      2. div-inv 35.6%

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

   5. Applied egg-rr 35.6%

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

      1. *-commutative 35.6%

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

      2. *-commutative 35.6%

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

      3. *-commutative 35.6%

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

      4. associate-/r* 35.6%

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

      5. metadata-eval 35.6%

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

   7. Simplified 35.6%

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

      1. *-commutative 35.6%

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

      2. sqrt-prod 99.5%

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

   9. Applied egg-rr 99.5%

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

   if -2.55000000000000006e-48 < b < 5.20000000000000004e-70

   1. Initial program 77.2%

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

      1. sqr-neg 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-*l* 77.2%

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

      4. *-commutative 77.2%

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

   3. Simplified 77.2%

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

      1. frac-2neg 77.2%

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

      2. div-inv 77.0%

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

   5. Applied egg-rr 80.9%

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

      1. *-commutative 80.9%

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

      2. *-commutative 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-/r* 80.9%

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

      5. metadata-eval 80.9%

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

   7. Simplified 80.9%

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

   if 5.20000000000000004e-70 < b

   1. Initial program 13.2%

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

      1. sqr-neg 13.2%

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

      2. sqr-neg 13.2%

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

      3. associate-*l* 14.1%

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

      4. *-commutative 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot -4} \cdot \sqrt{a}\right)\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 85.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -4}, b\right)\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+154)
   (/ (- b) a)
   (if (<= b -8.6e-44)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b -2.55e-48)
       (* (/ 1.0 (* a 2.0)) (fma (sqrt a) (sqrt (* c -4.0)) b))
       (if (<= b 3.7e-70)
         (* (/ -0.5 a) (- b (hypot b (sqrt (* a (* c -4.0))))))
         (/ (- c) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+154) {
		tmp = -b / a;
	} else if (b <= -8.6e-44) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= -2.55e-48) {
		tmp = (1.0 / (a * 2.0)) * fma(sqrt(a), sqrt((c * -4.0)), b);
	} else if (b <= 3.7e-70) {
		tmp = (-0.5 / a) * (b - hypot(b, sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -8.6e-44)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= -2.55e-48)
		tmp = Float64(Float64(1.0 / Float64(a * 2.0)) * fma(sqrt(a), sqrt(Float64(c * -4.0)), b));
	elseif (b <= 3.7e-70)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - hypot(b, sqrt(Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -8.6e-44], 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[LessEqual[b, -2.55e-48], N[(N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[a], $MachinePrecision] * N[Sqrt[N[(c * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.7e-70], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -5.00000000000000004e154

   1. Initial program 38.1%

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

      1. sqr-neg 38.1%

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

      2. sqr-neg 38.1%

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

      3. associate-*l* 38.1%

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

      4. *-commutative 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in b around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. mul-1-neg 97.7%

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

   6. Simplified 97.7%

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

   if -5.00000000000000004e154 < b < -8.60000000000000027e-44

   1. Initial program 97.2%

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

      1. sqr-neg 97.2%

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

      2. sqr-neg 97.2%

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

      3. associate-*l* 97.2%

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

      4. *-commutative 97.2%

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

   3. Simplified 97.2%

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

   if -8.60000000000000027e-44 < b < -2.55000000000000006e-48

   1. Initial program 3.3%

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

      1. sqr-neg 3.3%

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

      2. sqr-neg 3.3%

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

      3. associate-*l* 3.3%

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

      4. *-commutative 3.3%

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

   3. Simplified 3.3%

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

   4. Taylor expanded in b around 0 3.3%

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

      1. *-commutative 3.3%

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

      2. associate-*r* 3.3%

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

   6. Simplified 3.3%

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

      1. clear-num 3.3%

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

      2. inv-pow 3.3%

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

      3. *-commutative 3.3%

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

      4. *-un-lft-identity 3.3%

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

      5. times-frac 3.3%

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

      6. metadata-eval 3.3%

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

      7. +-commutative 3.3%

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

      8. add-sqr-sqrt 3.3%

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

      9. sqrt-unprod 3.3%

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

      10. sqr-neg 3.3%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 3.3%

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

   8. Applied egg-rr 3.3%

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

      1. unpow-1 3.3%

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

      2. associate-*r/ 3.3%

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

      3. *-commutative 3.3%

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

      4. +-commutative 3.3%

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

      5. *-commutative 3.3%

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

      6. associate-*l* 3.3%

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

   10. Simplified 3.3%

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

       1. expm1-log1p-u 3.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{a \cdot 2}{b + \sqrt{c \cdot \left(-4 \cdot a\right)}}}\right)\right)} \]

       2. expm1-udef 3.3%

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

       3. associate-/r/ 3.3%

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

       4. +-commutative 3.3%

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

       5. associate-*r* 3.3%

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

       6. *-commutative 3.3%

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

       7. sqrt-prod 98.0%

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

       8. fma-def 98.0%

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

   12. Applied egg-rr 98.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{a \cdot 2} \cdot \mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -4}, b\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 97.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{a \cdot 2} \cdot \mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -4}, b\right)\right)\right)} \]

       2. expm1-log1p 99.2%

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

   14. Simplified 99.2%

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

   if -2.55000000000000006e-48 < b < 3.7e-70

   1. Initial program 77.2%

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

      1. sqr-neg 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-*l* 77.2%

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

      4. *-commutative 77.2%

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

   3. Simplified 77.2%

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

      1. frac-2neg 77.2%

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

      2. div-inv 77.0%

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

   5. Applied egg-rr 80.9%

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

      1. *-commutative 80.9%

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

      2. *-commutative 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-/r* 80.9%

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

      5. metadata-eval 80.9%

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

   7. Simplified 80.9%

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

   if 3.7e-70 < b

   1. Initial program 13.2%

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

      1. sqr-neg 13.2%

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

      2. sqr-neg 13.2%

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

      3. associate-*l* 14.1%

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

      4. *-commutative 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \mathsf{fma}\left(\sqrt{a}, \sqrt{c \cdot -4}, b\right)\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \sqrt{a} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+154)
   (/ (- b) a)
   (if (<= b -8.6e-44)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b -2.55e-48)
       (/ (- (* (sqrt (* c -4.0)) (sqrt a)) b) (* a 2.0))
       (if (<= b 9e-70)
         (* (/ -0.5 a) (- b (hypot b (sqrt (* a (* c -4.0))))))
         (/ (- c) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+154) {
		tmp = -b / a;
	} else if (b <= -8.6e-44) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= -2.55e-48) {
		tmp = ((sqrt((c * -4.0)) * sqrt(a)) - b) / (a * 2.0);
	} else if (b <= 9e-70) {
		tmp = (-0.5 / a) * (b - hypot(b, sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+154) {
		tmp = -b / a;
	} else if (b <= -8.6e-44) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= -2.55e-48) {
		tmp = ((Math.sqrt((c * -4.0)) * Math.sqrt(a)) - b) / (a * 2.0);
	} else if (b <= 9e-70) {
		tmp = (-0.5 / a) * (b - Math.hypot(b, Math.sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+154:
		tmp = -b / a
	elif b <= -8.6e-44:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	elif b <= -2.55e-48:
		tmp = ((math.sqrt((c * -4.0)) * math.sqrt(a)) - b) / (a * 2.0)
	elif b <= 9e-70:
		tmp = (-0.5 / a) * (b - math.hypot(b, math.sqrt((a * (c * -4.0)))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -8.6e-44)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= -2.55e-48)
		tmp = Float64(Float64(Float64(sqrt(Float64(c * -4.0)) * sqrt(a)) - b) / Float64(a * 2.0));
	elseif (b <= 9e-70)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - hypot(b, sqrt(Float64(a * Float64(c * -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 <= -5e+154)
		tmp = -b / a;
	elseif (b <= -8.6e-44)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	elseif (b <= -2.55e-48)
		tmp = ((sqrt((c * -4.0)) * sqrt(a)) - b) / (a * 2.0);
	elseif (b <= 9e-70)
		tmp = (-0.5 / a) * (b - hypot(b, sqrt((a * (c * -4.0)))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -8.6e-44], 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[LessEqual[b, -2.55e-48], N[(N[(N[(N[Sqrt[N[(c * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9e-70], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -5.00000000000000004e154

   1. Initial program 38.1%

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

      1. sqr-neg 38.1%

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

      2. sqr-neg 38.1%

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

      3. associate-*l* 38.1%

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

      4. *-commutative 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in b around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. mul-1-neg 97.7%

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

   6. Simplified 97.7%

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

   if -5.00000000000000004e154 < b < -8.60000000000000027e-44

   1. Initial program 97.2%

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

      1. sqr-neg 97.2%

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

      2. sqr-neg 97.2%

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

      3. associate-*l* 97.2%

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

      4. *-commutative 97.2%

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

   3. Simplified 97.2%

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

   if -8.60000000000000027e-44 < b < -2.55000000000000006e-48

   1. Initial program 3.3%

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

      1. sqr-neg 3.3%

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

      2. sqr-neg 3.3%

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

      3. associate-*l* 3.3%

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

      4. *-commutative 3.3%

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

   3. Simplified 3.3%

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

   4. Taylor expanded in b around 0 3.3%

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

      1. *-commutative 3.3%

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

      2. associate-*r* 3.3%

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

   6. Simplified 3.3%

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

      1. *-commutative 3.3%

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

      2. sqrt-prod 99.2%

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

   8. Applied egg-rr 98.4%

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

   if -2.55000000000000006e-48 < b < 9.00000000000000044e-70

   1. Initial program 77.2%

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

      1. sqr-neg 77.2%

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

      2. sqr-neg 77.2%

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

      3. associate-*l* 77.2%

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

      4. *-commutative 77.2%

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

   3. Simplified 77.2%

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

      1. frac-2neg 77.2%

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

      2. div-inv 77.0%

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

   5. Applied egg-rr 80.9%

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

      1. *-commutative 80.9%

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

      2. *-commutative 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-/r* 80.9%

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

      5. metadata-eval 80.9%

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

   7. Simplified 80.9%

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

   if 9.00000000000000044e-70 < b

   1. Initial program 13.2%

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

      1. sqr-neg 13.2%

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

      2. sqr-neg 13.2%

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

      3. associate-*l* 14.1%

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

      4. *-commutative 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -2.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \sqrt{a} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-70}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -310000000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -310000000000.0)
   (- (/ c b) (/ b a))
   (if (<= b 1.2e-69)
     (* (/ -0.5 a) (- b (hypot b (sqrt (* a (* c -4.0))))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -310000000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.2e-69) {
		tmp = (-0.5 / a) * (b - hypot(b, sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -310000000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.2e-69) {
		tmp = (-0.5 / a) * (b - Math.hypot(b, Math.sqrt((a * (c * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -310000000000.0:
		tmp = (c / b) - (b / a)
	elif b <= 1.2e-69:
		tmp = (-0.5 / a) * (b - math.hypot(b, math.sqrt((a * (c * -4.0)))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -310000000000.0)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.2e-69)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - hypot(b, sqrt(Float64(a * Float64(c * -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 <= -310000000000.0)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.2e-69)
		tmp = (-0.5 / a) * (b - hypot(b, sqrt((a * (c * -4.0)))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -310000000000.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.2e-69], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1e11

   1. Initial program 63.7%

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

      1. sqr-neg 63.7%

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

      2. sqr-neg 63.7%

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

      3. associate-*l* 63.7%

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

      4. *-commutative 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in b around -inf 96.8%

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

      1. +-commutative 96.8%

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

      2. mul-1-neg 96.8%

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

      3. unsub-neg 96.8%

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

   6. Simplified 96.8%

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

   if -3.1e11 < b < 1.2000000000000001e-69

   1. Initial program 76.1%

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

      1. sqr-neg 76.1%

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

      2. sqr-neg 76.1%

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

      3. associate-*l* 76.1%

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

      4. *-commutative 76.1%

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

   3. Simplified 76.1%

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

      1. frac-2neg 76.1%

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

      2. div-inv 76.0%

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

   5. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. *-commutative 78.2%

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

      3. *-commutative 78.2%

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

      4. associate-/r* 78.2%

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

      5. metadata-eval 78.2%

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

   7. Simplified 78.2%

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

   if 1.2000000000000001e-69 < b

   1. Initial program 13.2%

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

      1. sqr-neg 13.2%

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

      2. sqr-neg 13.2%

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

      3. associate-*l* 14.1%

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

      4. *-commutative 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -310000000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (- b) a)
   (if (<= b 4.6e-70)
     (/ (- (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 <= -1e+153) {
		tmp = -b / a;
	} else if (b <= 4.6e-70) {
		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 <= (-1d+153)) then
        tmp = -b / a
    else if (b <= 4.6d-70) 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 <= -1e+153) {
		tmp = -b / a;
	} else if (b <= 4.6e-70) {
		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 <= -1e+153:
		tmp = -b / a
	elif b <= 4.6e-70:
		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 <= -1e+153)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.6e-70)
		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 <= -1e+153)
		tmp = -b / a;
	elseif (b <= 4.6e-70)
		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, -1e+153], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.6e-70], 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 < -1e153

   1. Initial program 38.1%

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

      1. sqr-neg 38.1%

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

      2. sqr-neg 38.1%

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

      3. associate-*l* 38.1%

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

      4. *-commutative 38.1%

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

   3. Simplified 38.1%

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

   4. Taylor expanded in b around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. mul-1-neg 97.7%

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

   6. Simplified 97.7%

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

   if -1e153 < b < 4.60000000000000001e-70

   1. Initial program 82.4%

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

      1. sqr-neg 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-*l* 82.4%

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

      4. *-commutative 82.4%

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

   3. Simplified 82.4%

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

   if 4.60000000000000001e-70 < b

   1. Initial program 13.2%

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

      1. sqr-neg 13.2%

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

      2. sqr-neg 13.2%

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

      3. associate-*l* 14.1%

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

      4. *-commutative 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-74)
   (- (/ c b) (/ b a))
   (if (<= b 3.4e-70)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-74) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.4e-70) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (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 <= (-6d-74)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.4d-70) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (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 <= -6e-74) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.4e-70) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-74:
		tmp = (c / b) - (b / a)
	elif b <= 3.4e-70:
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-74)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.4e-70)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / 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 <= -6e-74)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.4e-70)
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-74], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.4e-70], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.00000000000000014e-74

   1. Initial program 64.6%

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

      1. sqr-neg 64.6%

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

      2. sqr-neg 64.6%

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

      3. associate-*l* 64.6%

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

      4. *-commutative 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b around -inf 90.1%

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

      1. +-commutative 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

   6. Simplified 90.1%

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

   if -6.00000000000000014e-74 < b < 3.39999999999999995e-70

   1. Initial program 76.9%

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

      1. sqr-neg 76.9%

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

      2. sqr-neg 76.9%

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

      3. associate-*l* 76.9%

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

      4. *-commutative 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in b around 0 70.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]
   5. 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%

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

   6. Simplified 70.6%

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

      1. expm1-log1p-u 49.4%

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

      2. expm1-udef 18.4%

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

      3. +-commutative 18.4%

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

      4. add-sqr-sqrt 8.5%

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

      5. sqrt-unprod 18.6%

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

      6. sqr-neg 18.6%

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

      7. sqrt-unprod 9.9%

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

      8. add-sqr-sqrt 17.7%

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

   8. Applied egg-rr 17.7%

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

      1. expm1-def 48.1%

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

      2. expm1-log1p 68.2%

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

      3. +-commutative 68.2%

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

      4. *-commutative 68.2%

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

      5. associate-*l* 68.2%

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

   10. Simplified 68.2%

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

   if 3.39999999999999995e-70 < b

   1. Initial program 13.2%

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

      1. sqr-neg 13.2%

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

      2. sqr-neg 13.2%

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

      3. associate-*l* 14.1%

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

      4. *-commutative 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-74)
   (- (/ c b) (/ b a))
   (if (<= b 6e-70) (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-74) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6e-70) {
		tmp = (sqrt((a * (c * -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 <= (-2.8d-74)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6d-70) then
        tmp = (sqrt((a * (c * (-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 <= -2.8e-74) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6e-70) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-74:
		tmp = (c / b) - (b / a)
	elif b <= 6e-70:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-74)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6e-70)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -2.8e-74)
		tmp = (c / b) - (b / a);
	elseif (b <= 6e-70)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-74], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-70], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -2.79999999999999988e-74

   1. Initial program 64.6%

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

      1. sqr-neg 64.6%

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

      2. sqr-neg 64.6%

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

      3. associate-*l* 64.6%

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

      4. *-commutative 64.6%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b around -inf 90.1%

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

      1. +-commutative 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

   6. Simplified 90.1%

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

   if -2.79999999999999988e-74 < b < 6.0000000000000003e-70

   1. Initial program 76.9%

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

      1. sqr-neg 76.9%

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

      2. sqr-neg 76.9%

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

      3. associate-*l* 76.9%

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

      4. *-commutative 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in b around 0 70.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]
   5. 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%

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

   6. Simplified 70.6%

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

   if 6.0000000000000003e-70 < b

   1. Initial program 13.2%

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

      1. sqr-neg 13.2%

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

      2. sqr-neg 13.2%

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

      3. associate-*l* 14.1%

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

      4. *-commutative 14.1%

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

   3. Simplified 14.1%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 67.4% accurate, 12.8× 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 69.5%

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

      1. sqr-neg 69.5%

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

      2. sqr-neg 69.5%

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

      3. associate-*l* 69.5%

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

      4. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in b around -inf 71.6%

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

      1. +-commutative 71.6%

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

      2. mul-1-neg 71.6%

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

      3. unsub-neg 71.6%

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

   6. Simplified 71.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 28.4%

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

      1. sqr-neg 28.4%

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

      2. sqr-neg 28.4%

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

      3. associate-*l* 29.2%

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

      4. *-commutative 29.2%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in b around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   6. Simplified 69.7%

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

4. Final simplification 70.6%

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

Alternative 9: 42.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.02 \cdot 10^{-13}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2.02e-13) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.02e-13) {
		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 <= 2.02d-13) 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 <= 2.02e-13) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.02e-13:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.02e-13)
		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 <= 2.02e-13)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.02e-13], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.02e-13

   1. Initial program 68.6%

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

      1. sqr-neg 68.6%

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

      2. sqr-neg 68.6%

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

      3. associate-*l* 68.6%

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

      4. *-commutative 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in b around -inf 51.9%

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

      1. associate-*r/ 51.9%

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

      2. mul-1-neg 51.9%

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

   6. Simplified 51.9%

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

   if 2.02e-13 < b

   1. Initial program 10.2%

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

      1. sqr-neg 10.2%

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

      2. sqr-neg 10.2%

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

      3. associate-*l* 11.2%

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

      4. *-commutative 11.2%

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

   3. Simplified 11.2%

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

   4. Taylor expanded in b around -inf 2.5%

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

      1. +-commutative 2.5%

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

      2. mul-1-neg 2.5%

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

      3. unsub-neg 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in c around inf 35.1%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.02 \cdot 10^{-13}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 10: 67.2% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.5%

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

      1. sqr-neg 69.5%

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

      2. sqr-neg 69.5%

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

      3. associate-*l* 69.5%

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

      4. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in b around -inf 71.5%

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

      1. associate-*r/ 71.5%

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

      2. mul-1-neg 71.5%

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

   6. Simplified 71.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 28.4%

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

      1. sqr-neg 28.4%

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

      2. sqr-neg 28.4%

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

      3. associate-*l* 29.2%

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

      4. *-commutative 29.2%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in b around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   6. Simplified 69.7%

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

4. Final simplification 70.5%

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

Alternative 11: 10.5% 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 46.9%

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

   1. sqr-neg 46.9%

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

   2. sqr-neg 46.9%

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

   3. associate-*l* 47.3%

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

   4. *-commutative 47.3%

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

3. Simplified 47.3%

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

4. Taylor expanded in b around -inf 33.4%

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

   1. +-commutative 33.4%

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

   2. mul-1-neg 33.4%

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

   3. unsub-neg 33.4%

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

6. Simplified 33.4%

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

7. Taylor expanded in c around inf 14.8%

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

8. Final simplification 14.8%

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

Developer target: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* a (/ (- (- b) t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b < 0.0:
		tmp = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b < 0.0)
		tmp = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023275 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

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