jeff quadratic root 2

Percentage Accurate: 72.3% → 91.0%

Time: 20.4s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- 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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-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 = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = 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 = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Initial Program: 72.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- 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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-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 = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = 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 = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Alternative 1: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -3.7e+101)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b (+ b (* -2.0 (/ a (/ b c)))))))
       (* -0.5 (+ (* -2.0 (/ c b)) (* 2.0 (/ b a)))))
     (if (<= b 5.2e+128)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) b)) (/ (* b -2.0) (* a 2.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -3.7e+101) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + (b + (-2.0 * (a / (b / c))))));
		} else {
			tmp_2 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.2e+128) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (-b - b);
	} else {
		tmp_1 = (b * -2.0) / (a * 2.0);
	}
	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) - (c * (a * 4.0d0))))
    if (b <= (-3.7d+101)) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (b + (b + ((-2.0d0) * (a / (b / c))))))
        else
            tmp_2 = (-0.5d0) * (((-2.0d0) * (c / b)) + (2.0d0 * (b / a)))
        end if
        tmp_1 = tmp_2
    else if (b <= 5.2d+128) then
        if (b >= 0.0d0) then
            tmp_3 = (c * 2.0d0) / (-b - t_0)
        else
            tmp_3 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c * 2.0d0) / (-b - b)
    else
        tmp_1 = (b * (-2.0d0)) / (a * 2.0d0)
    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) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -3.7e+101) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + (b + (-2.0 * (a / (b / c))))));
		} else {
			tmp_2 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.2e+128) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (-b - b);
	} else {
		tmp_1 = (b * -2.0) / (a * 2.0);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -3.7e+101:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (b + (b + (-2.0 * (a / (b / c))))))
		else:
			tmp_2 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)))
		tmp_1 = tmp_2
	elif b <= 5.2e+128:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c * 2.0) / (-b - t_0)
		else:
			tmp_3 = (t_0 - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c * 2.0) / (-b - b)
	else:
		tmp_1 = (b * -2.0) / (a * 2.0)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -3.7e+101)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + Float64(b + Float64(-2.0 * Float64(a / Float64(b / c)))))));
		else
			tmp_2 = Float64(-0.5 * Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / a))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.2e+128)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	else
		tmp_1 = Float64(Float64(b * -2.0) / Float64(a * 2.0));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -3.7e+101)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (b + (b + (-2.0 * (a / (b / c))))));
		else
			tmp_3 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
		end
		tmp_2 = tmp_3;
	elseif (b <= 5.2e+128)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c * 2.0) / (-b - t_0);
		else
			tmp_4 = (t_0 - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c * 2.0) / (-b - b);
	else
		tmp_2 = (b * -2.0) / (a * 2.0);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3.7e+101], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[(b + N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.2e+128], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.6999999999999997e101

   1. Initial program 44.6%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in c around 0 44.9%

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

      1. associate-/l* 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in b around -inf 86.8%

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

   9. Taylor expanded in a around 0 96.4%

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

   if -3.6999999999999997e101 < b < 5.2e128

   1. Initial program 83.7%

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

   if 5.2e128 < b

   1. Initial program 51.5%

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

   3. Taylor expanded in b around -inf 51.5%

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

      1. *-commutative 51.5%

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

   5. Simplified 51.5%

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

   6. Taylor expanded in b around inf 51.6%

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

      1. sqrt-pow1 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow1 100.0%

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

      4. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{1 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot -2}{a \cdot 2}\\ \mathbf{if}\;b \leq 2.8 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* b -2.0) (* a 2.0))))
   (if (<= b 2.8e+116)
     (if (>= b 0.0)
       (/ (* c 2.0) (- (- b) (sqrt (- (* b b) (* c (* a 4.0))))))
       t_0)
     (if (>= b 0.0) (/ (* c 2.0) (- (- b) b)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= 2.8e+116) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * 2.0) / (-b - sqrt(((b * b) - (c * (a * 4.0)))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (-b - b);
	} else {
		tmp_1 = t_0;
	}
	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
    t_0 = (b * (-2.0d0)) / (a * 2.0d0)
    if (b <= 2.8d+116) then
        if (b >= 0.0d0) then
            tmp_2 = (c * 2.0d0) / (-b - sqrt(((b * b) - (c * (a * 4.0d0)))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c * 2.0d0) / (-b - b)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= 2.8e+116) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * 2.0) / (-b - Math.sqrt(((b * b) - (c * (a * 4.0)))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (-b - b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b * -2.0) / (a * 2.0)
	tmp_1 = 0
	if b <= 2.8e+116:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c * 2.0) / (-b - math.sqrt(((b * b) - (c * (a * 4.0)))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c * 2.0) / (-b - b)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * -2.0) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= 2.8e+116)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (b * -2.0) / (a * 2.0);
	tmp_2 = 0.0;
	if (b <= 2.8e+116)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c * 2.0) / (-b - sqrt(((b * b) - (c * (a * 4.0)))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c * 2.0) / (-b - b);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.8e+116], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.80000000000000004e116

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf 68.6%

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

      1. *-commutative 68.6%

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

   5. Simplified 68.6%

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

   if 2.80000000000000004e116 < b

   1. Initial program 51.5%

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

   3. Taylor expanded in b around -inf 51.5%

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

      1. *-commutative 51.5%

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

   5. Simplified 51.5%

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

   6. Taylor expanded in b around inf 51.6%

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

      1. sqrt-pow1 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow1 100.0%

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

      4. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{1 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot -2}{a \cdot 2}\\ \mathbf{if}\;b \leq 7.5 \cdot 10^{-131}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{\left(c \cdot a\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* b -2.0) (* a 2.0))))
   (if (<= b 7.5e-131)
     (if (>= b 0.0) (/ (* c 2.0) (- (- b) (sqrt (* (* c a) -4.0)))) t_0)
     (if (>= b 0.0) (/ (* c 2.0) (- (- b) b)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= 7.5e-131) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * 2.0) / (-b - sqrt(((c * a) * -4.0)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (-b - b);
	} else {
		tmp_1 = t_0;
	}
	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
    t_0 = (b * (-2.0d0)) / (a * 2.0d0)
    if (b <= 7.5d-131) then
        if (b >= 0.0d0) then
            tmp_2 = (c * 2.0d0) / (-b - sqrt(((c * a) * (-4.0d0))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c * 2.0d0) / (-b - b)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= 7.5e-131) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * 2.0) / (-b - Math.sqrt(((c * a) * -4.0)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (-b - b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b * -2.0) / (a * 2.0)
	tmp_1 = 0
	if b <= 7.5e-131:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c * 2.0) / (-b - math.sqrt(((c * a) * -4.0)))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c * 2.0) / (-b - b)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * -2.0) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= 7.5e-131)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - sqrt(Float64(Float64(c * a) * -4.0))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (b * -2.0) / (a * 2.0);
	tmp_2 = 0.0;
	if (b <= 7.5e-131)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c * 2.0) / (-b - sqrt(((c * a) * -4.0)));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c * 2.0) / (-b - b);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.5e-131], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.49999999999999964e-131

   1. Initial program 69.3%

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

   3. Taylor expanded in b around -inf 62.3%

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

      1. *-commutative 62.3%

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

   5. Simplified 62.3%

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

   6. Taylor expanded in b around 0 62.1%

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

      1. *-commutative 62.1%

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

   8. Simplified 62.1%

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

   if 7.49999999999999964e-131 < b

   1. Initial program 72.4%

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

   3. Taylor expanded in b around -inf 72.4%

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

      1. *-commutative 72.4%

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

   5. Simplified 72.4%

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

   6. Taylor expanded in b around inf 60.7%

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

      1. sqrt-pow1 84.9%

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

      2. metadata-eval 84.9%

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

      3. pow1 84.9%

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

      4. *-un-lft-identity 84.9%

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

   8. Applied egg-rr 84.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{1 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-131}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{\left(c \cdot a\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.0% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

3. Simplified 70.3%

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

5. Taylor expanded in c around 0 67.0%

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

   1. associate-/l* 67.4%

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

7. Simplified 67.4%

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

8. Taylor expanded in b around -inf 60.9%

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

9. Taylor expanded in a around 0 62.9%

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

10. Final simplification 62.9%

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

Alternative 5: 67.8% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

3. Taylor expanded in b around -inf 65.7%

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

   1. *-commutative 65.7%

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

5. Simplified 65.7%

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

6. Taylor expanded in b around inf 54.4%

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

   1. sqrt-pow1 62.8%

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

   2. metadata-eval 62.8%

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

   3. pow1 62.8%

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

   4. *-un-lft-identity 62.8%

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

8. Applied egg-rr 62.8%

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

9. Final simplification 62.8%

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

Alternative 6: 35.6% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = b / a
    else
        tmp = (b * (-2.0d0)) / (a * 2.0d0)
    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 * -2.0) / (a * 2.0);
	}
	return tmp;
}

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(b / a);
	else
		tmp = Float64(Float64(b * -2.0) / Float64(a * 2.0));
	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 * -2.0) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 70.3%

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

3. Taylor expanded in b around -inf 65.7%

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

   1. *-commutative 65.7%

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

5. Simplified 65.7%

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

6. Taylor expanded in b around inf 62.4%

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

7. Taylor expanded in c around inf 35.0%

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

8. Final simplification 35.0%

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

Reproduce

herbie shell --seed 2024011 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))