quadp (p42, positive)

Percentage Accurate: 53.8% → 85.1%

Time: 14.5s

Alternatives: 9

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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%

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

   2. Taylor expanded in b around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. mul-1-neg 97.7%

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

   4. Simplified 97.7%

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

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

   1. Initial program 97.1%

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

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

   1. Initial program 35.6%

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

      1. frac-2neg 35.6%

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

      2. div-inv 35.6%

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

   3. Applied egg-rr 35.6%

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

      1. associate-*r* 35.6%

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

      2. *-commutative 35.6%

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

      3. associate-*l* 35.6%

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

      4. *-commutative 35.6%

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

      5. associate-/r* 35.6%

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

      6. metadata-eval 35.6%

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

   5. Simplified 35.6%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot \frac{-0.5}{a}} \]
   6. 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%

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

   7. 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%

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

      1. frac-2neg 77.2%

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

      2. div-inv 77.0%

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

   3. Applied egg-rr 80.9%

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

      1. associate-*r* 80.9%

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

      2. *-commutative 80.9%

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

      3. associate-*l* 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-/r* 80.9%

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

      6. metadata-eval 80.9%

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

   5. 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 14.1%

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

   2. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   4. 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, 0.5× speedup

Math

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

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)
       (/ (- (* (sqrt (* c -4.0)) (sqrt a)) b) (* a 2.0))
       (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 = ((sqrt((c * -4.0)) * sqrt(a)) - b) / (a * 2.0);
	} else if (b <= 3.7e-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 <= 3.7e-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 <= 3.7e-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 <= 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

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 <= 3.7e-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, 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%

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

   2. Taylor expanded in b around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. mul-1-neg 97.7%

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

   4. Simplified 97.7%

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

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

   1. Initial program 97.2%

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

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

   1. Initial program 3.3%

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

   2. Taylor expanded in b around 0 3.3%

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

      1. associate-*r* 3.3%

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

      2. *-commutative 3.3%

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

      3. *-commutative 3.3%

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

   4. Simplified 3.3%

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

      1. add-sqr-sqrt 3.3%

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

      2. pow2 3.3%

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

      3. pow1/2 3.3%

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

      4. metadata-eval 3.3%

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

      5. sqrt-pow1 3.3%

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

      6. associate-*r* 3.3%

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

      7. metadata-eval 3.3%

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

      8. metadata-eval 3.3%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\left(c \cdot a\right) \cdot -4\right)}^{\color{blue}{0.25}}\right)}^{2}}{2 \cdot a} \]

   6. Applied egg-rr 3.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\left(c \cdot a\right) \cdot -4\right)}^{0.25}\right)}^{2}}}{2 \cdot a} \]
   7. Step-by-step derivation

      1. pow-pow 3.3%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\left(c \cdot a\right) \cdot -4\right)}^{\left(0.25 \cdot 2\right)}}}{2 \cdot a} \]

      2. *-commutative 3.3%

         \[\leadsto \frac{\left(-b\right) + {\left(\color{blue}{\left(a \cdot c\right)} \cdot -4\right)}^{\left(0.25 \cdot 2\right)}}{2 \cdot a} \]

      3. associate-*r* 3.3%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left(a \cdot \left(c \cdot -4\right)\right)}}^{\left(0.25 \cdot 2\right)}}{2 \cdot a} \]

      4. *-commutative 3.3%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left(\left(c \cdot -4\right) \cdot a\right)}}^{\left(0.25 \cdot 2\right)}}{2 \cdot a} \]

      5. metadata-eval 3.3%

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

      6. metadata-eval 3.3%

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

      7. unpow-prod-down 98.4%

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

      8. metadata-eval 98.4%

         \[\leadsto \frac{\left(-b\right) + {\left(c \cdot -4\right)}^{\color{blue}{0.5}} \cdot {a}^{\left(\sqrt{0.25}\right)}}{2 \cdot a} \]

      9. pow1/2 98.4%

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

      10. metadata-eval 98.4%

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

      11. pow1/2 98.4%

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

   8. Applied egg-rr 98.4%

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

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

   1. Initial program 77.2%

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

      1. frac-2neg 77.2%

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

      2. div-inv 77.0%

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

   3. Applied egg-rr 80.9%

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

      1. associate-*r* 80.9%

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

      2. *-commutative 80.9%

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

      3. associate-*l* 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-/r* 80.9%

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

      6. metadata-eval 80.9%

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

   5. 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 14.1%

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

   2. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 88.7%

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

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -275000000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -275000000000.0)
   (- (/ c b) (/ b a))
   (if (<= b 1.1e-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 <= -275000000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-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 <= -275000000000.0) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-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 <= -275000000000.0:
		tmp = (c / b) - (b / a)
	elif b <= 1.1e-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 <= -275000000000.0)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.1e-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 <= -275000000000.0)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.1e-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, -275000000000.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-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 < -2.75e11

   1. Initial program 63.7%

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

   2. Taylor expanded in b around -inf 96.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. 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%

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

   4. Simplified 96.8%

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

   if -2.75e11 < b < 1.1e-69

   1. Initial program 76.1%

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

      1. frac-2neg 76.1%

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

      2. div-inv 76.0%

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

   3. Applied egg-rr 78.2%

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

      1. associate-*r* 78.2%

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

      2. *-commutative 78.2%

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

      3. associate-*l* 78.2%

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

      4. *-commutative 78.2%

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

      5. associate-/r* 78.2%

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

      6. metadata-eval 78.2%

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

   5. 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 1.1e-69 < b

   1. Initial program 14.1%

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

   2. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -275000000000:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (- b) a)
   (if (<= b 5.5e-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 <= 5.5e-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 <= 5.5d-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 <= 5.5e-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 <= 5.5e-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 <= 5.5e-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 <= 5.5e-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, 5.5e-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%

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

   2. Taylor expanded in b around -inf 97.7%

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

      1. associate-*r/ 97.7%

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

      2. mul-1-neg 97.7%

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

   4. Simplified 97.7%

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

   if -1e153 < b < 5.5000000000000001e-70

   1. Initial program 82.4%

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

   if 5.5000000000000001e-70 < b

   1. Initial program 14.1%

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

   2. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.8e-69)
   (- (/ c b) (/ b a))
   (if (<= b 7.5e-70)
     (* (/ 0.5 a) (+ b (sqrt (* a (* c -4.0)))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.8e-69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.5e-70) {
		tmp = (0.5 / a) * (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.8d-69)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7.5d-70) then
        tmp = (0.5d0 / a) * (b + sqrt((a * (c * (-4.0d0)))))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.8e-69) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.5e-70) {
		tmp = (0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.8e-69:
		tmp = (c / b) - (b / a)
	elif b <= 7.5e-70:
		tmp = (0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.8e-69)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7.5e-70)
		tmp = Float64(Float64(0.5 / a) * Float64(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 <= -8.8e-69)
		tmp = (c / b) - (b / a);
	elseif (b <= 7.5e-70)
		tmp = (0.5 / a) * (b + sqrt((a * (c * -4.0))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.8e-69], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-70], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.8000000000000001e-69

   1. Initial program 64.6%

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

   2. Taylor expanded in b around -inf 90.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. 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%

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

   4. Simplified 90.1%

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

   if -8.8000000000000001e-69 < b < 7.49999999999999973e-70

   1. Initial program 76.9%

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

   2. Taylor expanded in b around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

      3. *-commutative 70.6%

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

   4. Simplified 70.6%

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

      1. frac-2neg 70.6%

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

      2. div-inv 70.4%

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

      3. +-commutative 70.4%

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

      4. associate-*r* 70.4%

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

      5. add-sqr-sqrt 33.6%

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

      6. sqrt-unprod 69.8%

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

      7. sqr-neg 69.8%

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

      8. sqrt-unprod 36.8%

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

      9. add-sqr-sqrt 68.1%

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

      10. *-commutative 68.1%

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

      11. distribute-rgt-neg-in 68.1%

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

      12. metadata-eval 68.1%

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

      13. metadata-eval 68.1%

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

      14. div-inv 68.1%

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

      15. clear-num 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. *-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. associate-*l* 68.1%

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

   8. Simplified 68.1%

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

      1. expm1-log1p-u 48.1%

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

      2. expm1-udef 17.7%

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

      3. distribute-neg-in 17.7%

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

      4. add-cube-cbrt 17.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{a} \cdot \left(\color{blue}{\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b}} + \left(-\sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)} - 1 \]

      5. fma-def 17.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{a} \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}, \sqrt[3]{-b}, -\sqrt{c \cdot \left(a \cdot -4\right)}\right)}\right)} - 1 \]

      6. fma-neg 17.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-0.5}{a} \cdot \color{blue}{\left(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \sqrt[3]{-b} - \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\right)} - 1 \]

      7. add-cube-cbrt 17.7%

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

      8. associate-*r* 17.7%

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

      9. *-commutative 17.7%

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

      10. associate-*r* 17.7%

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

   10. Applied egg-rr 17.7%

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

       1. expm1-def 48.1%

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

       2. expm1-log1p 68.1%

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

       3. sub-neg 68.1%

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

       4. distribute-neg-in 68.1%

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

       5. distribute-rgt-neg-in 68.1%

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

       6. distribute-lft-neg-in 68.1%

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

       7. distribute-neg-frac 68.1%

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

       8. metadata-eval 68.1%

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

   12. Simplified 68.1%

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

   if 7.49999999999999973e-70 < b

   1. Initial program 14.1%

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

   2. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-68}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-68)
   (- (/ c b) (/ b a))
   (if (<= b 3.8e-70)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-68) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.8e-70) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d-68)) then
        tmp = (c / b) - (b / a)
    else if (b <= 3.8d-70) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-68) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.8e-70) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-68:
		tmp = (c / b) - (b / a)
	elif b <= 3.8e-70:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-68)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 3.8e-70)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-68)
		tmp = (c / b) - (b / a);
	elseif (b <= 3.8e-70)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-68], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.8e-70], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3e-68

   1. Initial program 64.6%

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

   2. Taylor expanded in b around -inf 90.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. 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%

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

   4. Simplified 90.1%

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

   if -3e-68 < b < 3.7999999999999998e-70

   1. Initial program 76.9%

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

      1. add-sqr-sqrt 76.5%

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

      2. pow2 76.5%

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

      3. pow1/2 76.5%

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

      4. sqrt-pow1 76.6%

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

      5. fma-neg 76.6%

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

      6. distribute-lft-neg-in 76.6%

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

      7. associate-*r* 76.6%

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

      8. metadata-eval 76.6%

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

      9. metadata-eval 76.6%

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

   3. Applied egg-rr 76.6%

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

   4. Taylor expanded in a around inf 34.7%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-4 \cdot c\right) + -1 \cdot \log \left(\frac{1}{a}\right)\right)}\right)}^{2} - b}}{2 \cdot a} \]

   6. Simplified 70.6%

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

   if 3.7999999999999998e-70 < b

   1. Initial program 14.1%

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

   2. Taylor expanded in b around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-68}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 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%

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

   2. Taylor expanded in b around -inf 71.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. 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%

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

   4. Simplified 71.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 29.2%

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

   2. Taylor expanded in b around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   4. 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 8: 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%

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

   2. Taylor expanded in b around -inf 71.5%

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

      1. associate-*r/ 71.5%

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

      2. mul-1-neg 71.5%

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

   4. Simplified 71.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 29.2%

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

   2. Taylor expanded in b around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   4. 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 9: 35.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.3%

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

2. Taylor expanded in b around -inf 33.6%

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

   1. associate-*r/ 33.6%

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

   2. mul-1-neg 33.6%

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

4. Simplified 33.6%

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

5. Final simplification 33.6%

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

Developer target: 71.5% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $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 "quadp (p42, positive)"
  :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)))