jeff quadratic root 1

Percentage Accurate: 72.6% → 90.0%

Time: 19.3s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array} \end{array} \]

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)) (/ (* 2.0 c) (+ (- b) t_0)))))

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 = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: 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 = (2.0d0 * c) / (-b + t_0)
    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 = (2.0 * c) / (-b + t_0);
	}
	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 = (2.0 * c) / (-b + t_0)
	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(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	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 = (2.0 * c) / (-b + t_0);
	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[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array} \end{array} \]

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)) (/ (* 2.0 c) (+ (- b) t_0)))))

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 = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: 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 = (2.0d0 * c) / (-b + t_0)
    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 = (2.0 * c) / (-b + t_0);
	}
	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 = (2.0 * c) / (-b + t_0)
	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(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	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 = (2.0 * c) / (-b + t_0);
	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[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + t\_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* a (* c -4.0))))))
   (if (<= b -1.1e+61)
     (if (>= b 0.0) (* -0.5 (/ 1.0 (* (/ 0.5 b) a))) (/ (* c 2.0) (- (- b) b)))
     (if (<= b 4.6e+57)
       (if (>= b 0.0) (* -0.5 (/ (+ b t_0) a)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (/ (- b) a) (/ b a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (a * (c * -4.0))));
	double tmp_1;
	if (b <= -1.1e+61) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (1.0 / ((0.5 / b) * a));
		} else {
			tmp_2 = (c * 2.0) / (-b - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.6e+57) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * ((b + t_0) / a);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -b / a;
	} else {
		tmp_1 = b / a;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(a * Float64(c * -4.0))))
	tmp_1 = 0.0
	if (b <= -1.1e+61)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(1.0 / Float64(Float64(0.5 / b) * a)));
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.6e+57)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 * Float64(Float64(b + t_0) / a));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-b) / a);
	else
		tmp_1 = Float64(b / a);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.1e+61], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(1.0 / N[(N[(0.5 / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.6e+57], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(b / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1e61

   1. Initial program 52.7%

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

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}\\ } \end{array}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 94.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{-1 \cdot b - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. clear-num 94.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      2. inv-pow 94.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      3. fma-udef 94.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      4. add-sqr-sqrt 94.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      5. hypot-def 94.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   7. Applied egg-rr 94.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   8. Step-by-step derivation

      1. unpow-1 94.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   9. Simplified 94.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   10. Taylor expanded in a around 0 94.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   11. Step-by-step derivation

       1. associate-*r/ 94.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5 \cdot a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       2. *-rgt-identity 94.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5 \cdot a}{\color{blue}{b \cdot 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       3. times-frac 94.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{b} \cdot \frac{a}{1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       4. rem-square-sqrt 94.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       5. associate-*r/ 94.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \color{blue}{\left(\sqrt{a} \cdot \frac{\sqrt{a}}{1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       6. /-rgt-identity 94.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \left(\sqrt{a} \cdot \color{blue}{\sqrt{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       7. rem-square-sqrt 94.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   12. Simplified 94.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{b} \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   if -1.1e61 < b < 4.5999999999999998e57

   1. Initial program 85.9%

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

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}\\ } \end{array}} \]
   4. Add Preprocessing

   if 4.5999999999999998e57 < b

   1. Initial program 66.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. sqr-neg 66.2%

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

      2. sqr-neg 66.2%

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

      3. associate-*l* 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-/l* 66.2%

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

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 94.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r/ 94.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

      2. mul-1-neg 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in b around inf 94.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{-2 \cdot \frac{a}{b}}}\\ \end{array} \]
   9. Step-by-step derivation

      1. associate-*r/ 94.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]

   10. Simplified 94.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]
   11. Step-by-step derivation

       1. frac-2neg 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

       2. metadata-eval 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

       3. div-inv 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

       4. add-sqr-sqrt 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{b} \cdot \sqrt{b}}}\\ \end{array} \]

       5. sqrt-unprod 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{b \cdot b}}}\\ \end{array} \]

       6. sqr-neg 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\\ \end{array} \]

       7. sqrt-unprod 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{-b} \cdot \sqrt{-b}}}\\ \end{array} \]

       8. add-sqr-sqrt 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{-b}}\\ \end{array} \]

       9. distribute-frac-neg 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{1}{\frac{--2 \cdot a}{-b}}}\\ \end{array} \]

       10. frac-2neg 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{1}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]

       11. clear-num 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{b}{-2 \cdot a}}\\ \end{array} \]

       12. *-un-lft-identity 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2} \cdot \frac{1 \cdot b}{-2 \cdot a}\\ \end{array} \]

       13. times-frac 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2 \cdot \left(\frac{1}{-2} \cdot \frac{b}{a}\right)}\\ \end{array} \]

       14. metadata-eval 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2} \cdot \left(-0.5 \cdot \frac{b}{a}\right)\\ \end{array} \]

   12. Applied egg-rr 94.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \frac{b}{a}\right)\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r* 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot -0.5\right) \cdot \frac{b}{a}\\ \end{array} \]

       2. metadata-eval 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{b}{a}\\ \end{array} \]

       3. *-lft-identity 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

   14. Simplified 94.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+35}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_0 - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (<= b -2.8e+35)
     (if (>= b 0.0) (* -0.5 (/ 1.0 (* (/ 0.5 b) a))) (/ (* c 2.0) (- (- b) b)))
     (if (<= b 4.6e+57)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ 2.0 (/ (- t_0 b) c)))
       (if (>= b 0.0) (/ (- b) a) (/ b a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp_1;
	if (b <= -2.8e+35) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (1.0 / ((0.5 / b) * a));
		} else {
			tmp_2 = (c * 2.0) / (-b - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.6e+57) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -b / a;
	} else {
		tmp_1 = b / a;
	}
	return tmp_1;
}

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
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b <= (-2.8d+35)) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * (1.0d0 / ((0.5d0 / b) * a))
        else
            tmp_2 = (c * 2.0d0) / (-b - b)
        end if
        tmp_1 = tmp_2
    else if (b <= 4.6d+57) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 / ((t_0 - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = -b / a
    else
        tmp_1 = b / a
    end if
    code = tmp_1
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_1;
	if (b <= -2.8e+35) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (1.0 / ((0.5 / b) * a));
		} else {
			tmp_2 = (c * 2.0) / (-b - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.6e+57) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -b / a;
	} else {
		tmp_1 = b / a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp_1 = 0
	if b <= -2.8e+35:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * (1.0 / ((0.5 / b) * a))
		else:
			tmp_2 = (c * 2.0) / (-b - b)
		tmp_1 = tmp_2
	elif b <= 4.6e+57:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (a * 2.0)
		else:
			tmp_3 = 2.0 / ((t_0 - b) / c)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -b / a
	else:
		tmp_1 = b / a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp_1 = 0.0
	if (b <= -2.8e+35)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(1.0 / Float64(Float64(0.5 / b) * a)));
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.6e+57)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(t_0 - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-b) / a);
	else
		tmp_1 = Float64(b / a);
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp_2 = 0.0;
	if (b <= -2.8e+35)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * (1.0 / ((0.5 / b) * a));
		else
			tmp_3 = (c * 2.0) / (-b - b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 4.6e+57)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = 2.0 / ((t_0 - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -b / a;
	else
		tmp_2 = b / a;
	end
	tmp_5 = tmp_2;
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[LessEqual[b, -2.8e+35], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(1.0 / N[(N[(0.5 / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.6e+57], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$0 - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(b / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.79999999999999999e35

   1. Initial program 55.4%

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

   3. Simplified 55.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}\\ } \end{array}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 94.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{-1 \cdot b - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. clear-num 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      2. inv-pow 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      3. fma-udef 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      4. add-sqr-sqrt 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      5. hypot-def 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   7. Applied egg-rr 94.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   8. Step-by-step derivation

      1. unpow-1 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   9. Simplified 94.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   10. Taylor expanded in a around 0 94.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   11. Step-by-step derivation

       1. associate-*r/ 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5 \cdot a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       2. *-rgt-identity 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5 \cdot a}{\color{blue}{b \cdot 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       3. times-frac 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{b} \cdot \frac{a}{1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       4. rem-square-sqrt 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       5. associate-*r/ 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \color{blue}{\left(\sqrt{a} \cdot \frac{\sqrt{a}}{1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       6. /-rgt-identity 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \left(\sqrt{a} \cdot \color{blue}{\sqrt{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       7. rem-square-sqrt 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   12. Simplified 94.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{b} \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   if -2.79999999999999999e35 < b < 4.5999999999999998e57

   1. Initial program 85.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. sqr-neg 85.4%

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

      2. sqr-neg 85.4%

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

      3. associate-*l* 85.5%

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

      4. *-commutative 85.5%

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

      5. associate-/l* 85.5%

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

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]
   4. Add Preprocessing

   if 4.5999999999999998e57 < b

   1. Initial program 66.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. sqr-neg 66.2%

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

      2. sqr-neg 66.2%

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

      3. associate-*l* 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-/l* 66.2%

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

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 94.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r/ 94.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

      2. mul-1-neg 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in b around inf 94.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{-2 \cdot \frac{a}{b}}}\\ \end{array} \]
   9. Step-by-step derivation

      1. associate-*r/ 94.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]

   10. Simplified 94.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]
   11. Step-by-step derivation

       1. frac-2neg 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

       2. metadata-eval 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

       3. div-inv 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

       4. add-sqr-sqrt 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{b} \cdot \sqrt{b}}}\\ \end{array} \]

       5. sqrt-unprod 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{b \cdot b}}}\\ \end{array} \]

       6. sqr-neg 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\\ \end{array} \]

       7. sqrt-unprod 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{-b} \cdot \sqrt{-b}}}\\ \end{array} \]

       8. add-sqr-sqrt 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{-b}}\\ \end{array} \]

       9. distribute-frac-neg 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{1}{\frac{--2 \cdot a}{-b}}}\\ \end{array} \]

       10. frac-2neg 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{1}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]

       11. clear-num 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{b}{-2 \cdot a}}\\ \end{array} \]

       12. *-un-lft-identity 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2} \cdot \frac{1 \cdot b}{-2 \cdot a}\\ \end{array} \]

       13. times-frac 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2 \cdot \left(\frac{1}{-2} \cdot \frac{b}{a}\right)}\\ \end{array} \]

       14. metadata-eval 94.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2} \cdot \left(-0.5 \cdot \frac{b}{a}\right)\\ \end{array} \]

   12. Applied egg-rr 94.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \frac{b}{a}\right)\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r* 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot -0.5\right) \cdot \frac{b}{a}\\ \end{array} \]

       2. metadata-eval 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{b}{a}\\ \end{array} \]

       3. *-lft-identity 94.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

   14. Simplified 94.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+35}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+35}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+35)
   (if (>= b 0.0) (* -0.5 (/ 1.0 (* (/ 0.5 b) a))) (/ (* c 2.0) (- (- b) b)))
   (if (>= b 0.0)
     (/ (- b) a)
     (/ 2.0 (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) c)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.8e+35) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (1.0 / ((0.5 / b) * a));
		} else {
			tmp_2 = (c * 2.0) / (-b - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -b / a;
	} else {
		tmp_1 = 2.0 / ((sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
	}
	return tmp_1;
}

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
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-2.8d+35)) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * (1.0d0 / ((0.5d0 / b) * a))
        else
            tmp_2 = (c * 2.0d0) / (-b - b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = -b / a
    else
        tmp_1 = 2.0d0 / ((sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / c)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.8e+35) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (1.0 / ((0.5 / b) * a));
		} else {
			tmp_2 = (c * 2.0) / (-b - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -b / a;
	} else {
		tmp_1 = 2.0 / ((Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -2.8e+35:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * (1.0 / ((0.5 / b) * a))
		else:
			tmp_2 = (c * 2.0) / (-b - b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -b / a
	else:
		tmp_1 = 2.0 / ((math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / c)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -2.8e+35)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(1.0 / Float64(Float64(0.5 / b) * a)));
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-b) / a);
	else
		tmp_1 = Float64(2.0 / Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / c));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -2.8e+35)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * (1.0 / ((0.5 / b) * a));
		else
			tmp_3 = (c * 2.0) / (-b - b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -b / a;
	else
		tmp_2 = 2.0 / ((sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+35], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(1.0 / N[(N[(0.5 / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(2.0 / N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.79999999999999999e35

   1. Initial program 55.4%

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

   3. Simplified 55.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}\\ } \end{array}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 94.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{-1 \cdot b - b}\\ \end{array} \]
   6. Step-by-step derivation

      1. clear-num 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      2. inv-pow 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      3. fma-udef 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      4. add-sqr-sqrt 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

      5. hypot-def 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   7. Applied egg-rr 94.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   8. Step-by-step derivation

      1. unpow-1 94.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   9. Simplified 94.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   10. Taylor expanded in a around 0 94.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   11. Step-by-step derivation

       1. associate-*r/ 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5 \cdot a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       2. *-rgt-identity 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5 \cdot a}{\color{blue}{b \cdot 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       3. times-frac 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{b} \cdot \frac{a}{1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       4. rem-square-sqrt 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       5. associate-*r/ 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \color{blue}{\left(\sqrt{a} \cdot \frac{\sqrt{a}}{1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       6. /-rgt-identity 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \left(\sqrt{a} \cdot \color{blue}{\sqrt{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

       7. rem-square-sqrt 94.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   12. Simplified 94.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{b} \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   if -2.79999999999999999e35 < b

   1. Initial program 78.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. sqr-neg 78.1%

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

      2. sqr-neg 78.1%

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

      3. associate-*l* 78.1%

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

      4. *-commutative 78.1%

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

      5. associate-/l* 78.1%

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

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 75.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r/ 75.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

      2. mul-1-neg 75.6%

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

   7. Simplified 75.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+35}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* -0.5 (+ (* -2.0 (/ c b)) (* 2.0 (/ b a))))
   (/ (* c 2.0) (- (- b) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	} else {
		tmp = (c * 2.0) / (-b - 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 >= 0.0d0) then
        tmp = (-0.5d0) * (((-2.0d0) * (c / b)) + (2.0d0 * (b / a)))
    else
        tmp = (c * 2.0d0) / (-b - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	} else {
		tmp = (c * 2.0) / (-b - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)))
	else:
		tmp = (c * 2.0) / (-b - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / a))));
	else
		tmp = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	else
		tmp = (c * 2.0) / (-b - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.9%

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

3. Simplified 72.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}\\ } \end{array}} \]
4. Add Preprocessing

5. Taylor expanded in b around -inf 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{-1 \cdot b - b}\\ \end{array} \]

6. Taylor expanded in b around inf 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

7. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
8. Add Preprocessing

Alternative 5: 67.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -0.5 (/ 1.0 (* (/ 0.5 b) a))) (/ (* c 2.0) (- (- b) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (1.0 / ((0.5 / b) * a));
	} else {
		tmp = (c * 2.0) / (-b - 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 >= 0.0d0) then
        tmp = (-0.5d0) * (1.0d0 / ((0.5d0 / b) * a))
    else
        tmp = (c * 2.0d0) / (-b - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (1.0 / ((0.5 / b) * a));
	} else {
		tmp = (c * 2.0) / (-b - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -0.5 * (1.0 / ((0.5 / b) * a))
	else:
		tmp = (c * 2.0) / (-b - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(1.0 / Float64(Float64(0.5 / b) * a)));
	else
		tmp = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -0.5 * (1.0 / ((0.5 / b) * a));
	else
		tmp = (c * 2.0) / (-b - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-0.5 * N[(1.0 / N[(N[(0.5 / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.9%

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

3. Simplified 72.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}\\ } \end{array}} \]
4. Add Preprocessing

5. Taylor expanded in b around -inf 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{-1 \cdot b - b}\\ \end{array} \]
6. Step-by-step derivation

   1. clear-num 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   2. inv-pow 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   3. fma-udef 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   4. add-sqr-sqrt 61.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   5. hypot-def 62.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

7. Applied egg-rr 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
8. Step-by-step derivation

   1. unpow-1 62.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

9. Simplified 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

10. Taylor expanded in a around 0 69.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
11. Step-by-step derivation

    1. associate-*r/ 69.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5 \cdot a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

    2. *-rgt-identity 69.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5 \cdot a}{\color{blue}{b \cdot 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

    3. times-frac 69.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{b} \cdot \frac{a}{1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

    4. rem-square-sqrt 49.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

    5. associate-*r/ 49.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \color{blue}{\left(\sqrt{a} \cdot \frac{\sqrt{a}}{1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

    6. /-rgt-identity 49.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \left(\sqrt{a} \cdot \color{blue}{\sqrt{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

    7. rem-square-sqrt 69.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot \color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

12. Simplified 69.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\color{blue}{\frac{0.5}{b} \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

13. Final simplification 69.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{0.5}{b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
14. Add Preprocessing

Alternative 6: 67.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{b \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -0.5 (/ 1.0 (/ a (* b 2.0)))) (/ (* c 2.0) (- (- b) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (1.0 / (a / (b * 2.0)));
	} else {
		tmp = (c * 2.0) / (-b - 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 >= 0.0d0) then
        tmp = (-0.5d0) * (1.0d0 / (a / (b * 2.0d0)))
    else
        tmp = (c * 2.0d0) / (-b - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (1.0 / (a / (b * 2.0)));
	} else {
		tmp = (c * 2.0) / (-b - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -0.5 * (1.0 / (a / (b * 2.0)))
	else:
		tmp = (c * 2.0) / (-b - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(1.0 / Float64(a / Float64(b * 2.0))));
	else
		tmp = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -0.5 * (1.0 / (a / (b * 2.0)));
	else
		tmp = (c * 2.0) / (-b - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-0.5 * N[(1.0 / N[(a / N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.9%

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

3. Simplified 72.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}\\ } \end{array}} \]
4. Add Preprocessing

5. Taylor expanded in b around -inf 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{-1 \cdot b - b}\\ \end{array} \]
6. Step-by-step derivation

   1. clear-num 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   2. inv-pow 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   3. fma-udef 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   4. add-sqr-sqrt 61.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

   5. hypot-def 62.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot {\left(\frac{a}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

7. Applied egg-rr 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{{\left(\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
8. Step-by-step derivation

   1. unpow-1 62.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

9. Simplified 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

10. Taylor expanded in b around inf 69.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{\color{blue}{2 \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
11. Step-by-step derivation

    1. *-commutative 69.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{\color{blue}{b \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

12. Simplified 69.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{\color{blue}{b \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]

13. Final simplification 69.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{1}{\frac{a}{b \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
14. Add Preprocessing

Alternative 7: 67.3% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-2 \cdot \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (/ (- b) a) (/ 2.0 (* -2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -b / a;
	} else {
		tmp = 2.0 / (-2.0 * (b / c));
	}
	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 >= 0.0d0) then
        tmp = -b / a
    else
        tmp = 2.0d0 / ((-2.0d0) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -b / a;
	} else {
		tmp = 2.0 / (-2.0 * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -b / a
	else:
		tmp = 2.0 / (-2.0 * (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(2.0 / Float64(-2.0 * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -b / a;
	else
		tmp = 2.0 / (-2.0 * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(2.0 / N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.9%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
2. Step-by-step derivation

   1. sqr-neg 71.9%

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

   2. sqr-neg 71.9%

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

   3. associate-*l* 71.9%

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

   4. *-commutative 71.9%

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

   5. associate-/l* 71.8%

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

3. Simplified 71.8%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
6. Step-by-step derivation

   1. associate-*r/ 69.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   2. mul-1-neg 69.9%

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

7. Simplified 69.9%

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

8. Taylor expanded in b around -inf 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{-2 \cdot \frac{b}{c}}}\\ \end{array} \]
9. Step-by-step derivation

   1. *-commutative 69.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{b}{c} \cdot -2}}\\ \end{array} \]

10. Simplified 69.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{b}{c} \cdot -2}}\\ \end{array} \]

11. Final simplification 69.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-2 \cdot \frac{b}{c}}\\ \end{array} \]
12. Add Preprocessing

Alternative 8: 35.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (>= b 0.0) (/ (- b) a) (/ b a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -b / a;
	} else {
		tmp = b / 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) :: tmp
    if (b >= 0.0d0) then
        tmp = -b / a
    else
        tmp = b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -b / a;
	} else {
		tmp = b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -b / a
	else:
		tmp = b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(b / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -b / a;
	else
		tmp = b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(b / a), $MachinePrecision]]

Derivation

1. Initial program 71.9%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
2. Step-by-step derivation

   1. sqr-neg 71.9%

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

   2. sqr-neg 71.9%

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

   3. associate-*l* 71.9%

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

   4. *-commutative 71.9%

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

   5. associate-/l* 71.8%

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

3. Simplified 71.8%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
6. Step-by-step derivation

   1. associate-*r/ 69.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   2. mul-1-neg 69.9%

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

7. Simplified 69.9%

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

8. Taylor expanded in b around inf 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{-2 \cdot \frac{a}{b}}}\\ \end{array} \]
9. Step-by-step derivation

   1. associate-*r/ 37.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]

10. Simplified 37.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]
11. Step-by-step derivation

    1. frac-2neg 37.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

    2. metadata-eval 37.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

    3. div-inv 37.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{b}}\\ \end{array} \]

    4. add-sqr-sqrt 35.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{b} \cdot \sqrt{b}}}\\ \end{array} \]

    5. sqrt-unprod 37.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{b \cdot b}}}\\ \end{array} \]

    6. sqr-neg 37.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\\ \end{array} \]

    7. sqrt-unprod 37.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{\sqrt{-b} \cdot \sqrt{-b}}}\\ \end{array} \]

    8. add-sqr-sqrt 37.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{-\frac{-2 \cdot a}{-b}}\\ \end{array} \]

    9. distribute-frac-neg 37.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{1}{\frac{--2 \cdot a}{-b}}}\\ \end{array} \]

    10. frac-2neg 37.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{1}{\frac{-2 \cdot a}{b}}}\\ \end{array} \]

    11. clear-num 37.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{b}{-2 \cdot a}}\\ \end{array} \]

    12. *-un-lft-identity 37.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2} \cdot \frac{1 \cdot b}{-2 \cdot a}\\ \end{array} \]

    13. times-frac 37.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2 \cdot \left(\frac{1}{-2} \cdot \frac{b}{a}\right)}\\ \end{array} \]

    14. metadata-eval 37.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{-2} \cdot \left(-0.5 \cdot \frac{b}{a}\right)\\ \end{array} \]

12. Applied egg-rr 37.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(-0.5 \cdot \frac{b}{a}\right)\\ \end{array} \]
13. Step-by-step derivation

    1. associate-*r* 37.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot -0.5\right) \cdot \frac{b}{a}\\ \end{array} \]

    2. metadata-eval 37.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{b}{a}\\ \end{array} \]

    3. *-lft-identity 37.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

14. Simplified 37.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

15. Final simplification 37.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))