jeff quadratic root 1

Percentage Accurate: 72.4% → 90.6%

Time: 21.8s

Alternatives: 5

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: 72.4% 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: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -4.8 \cdot 10^{+121}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))) (t_1 (/ (- c) b)))
   (if (<= b -4.8e+121)
     (if (>= b 0.0) (/ (- b) a) t_1)
     (if (<= b 1.5e+136)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (* -0.5 (+ (* -2.0 (/ c b)) (* 2.0 (/ b a)))) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -4.8e+121) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.5e+136) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	} else {
		tmp_1 = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    t_1 = -c / b
    if (b <= (-4.8d+121)) then
        if (b >= 0.0d0) then
            tmp_2 = -b / a
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= 1.5d+136) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (-0.5d0) * (((-2.0d0) * (c / b)) + (2.0d0 * (b / a)))
    else
        tmp_1 = t_1
    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 t_1 = -c / b;
	double tmp_1;
	if (b <= -4.8e+121) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.5e+136) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	t_1 = -c / b
	tmp_1 = 0
	if b <= -4.8e+121:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -b / a
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= 1.5e+136:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (a * 2.0)
		else:
			tmp_3 = (c * 2.0) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)))
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_1 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -4.8e+121)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.5e+136)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-0.5 * Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / a))));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	t_1 = -c / b;
	tmp_2 = 0.0;
	if (b <= -4.8e+121)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -b / a;
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.5e+136)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = (c * 2.0) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	else
		tmp_2 = t_1;
	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]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -4.8e+121], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], t$95$1], If[LessEqual[b, 1.5e+136], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], 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], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.8e121

   1. Initial program 42.5%

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

   3. Simplified 42.5%

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

   4. Taylor expanded in b around -inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. distribute-neg-frac 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in b around inf 95.7%

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

   8. Taylor expanded in c around 0 95.7%

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

      1. associate-*r/ 95.7%

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

      2. associate-/l* 95.7%

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

      3. associate-/r/ 95.7%

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

   10. Simplified 95.7%

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

   11. Taylor expanded in a around 0 95.7%

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

       1. associate-*r/ 95.7%

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

       2. mul-1-neg 95.7%

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

   13. Simplified 95.7%

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

   if -4.8e121 < b < 1.49999999999999989e136

   1. Initial program 90.8%

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

   if 1.49999999999999989e136 < b

   1. Initial program 44.4%

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

   3. Simplified 44.4%

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

   4. Taylor expanded in b around -inf 44.4%

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

      1. mul-1-neg 44.4%

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

      2. distribute-neg-frac 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in b around inf 98.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+121}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(c \cdot -4\right)}\\ t_1 := -0.5 \cdot \frac{b + b}{a}\\ t_2 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(\frac{a \cdot 2}{\frac{b}{c}} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-8}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* c -4.0))))
        (t_1 (* -0.5 (/ (+ b b) a)))
        (t_2 (/ (- c) b)))
   (if (<= b -2.2e-43)
     (if (>= b 0.0) t_1 (* c (/ 2.0 (- (- (/ (* a 2.0) (/ b c)) b) b))))
     (if (<= b -5e-310)
       (if (>= b 0.0) t_1 (* c (/ 2.0 (- t_0 b))))
       (if (<= b 1e-8)
         (if (>= b 0.0) (* -0.5 (/ (+ b t_0) a)) t_2)
         (if (>= b 0.0) (* -0.5 (+ (* -2.0 (/ c b)) (* 2.0 (/ b a)))) t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (c * -4.0)));
	double t_1 = -0.5 * ((b + b) / a);
	double t_2 = -c / b;
	double tmp_1;
	if (b <= -2.2e-43) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = c * (2.0 / ((((a * 2.0) / (b / c)) - b) - b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = c * (2.0 / (t_0 - b));
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e-8) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * ((b + t_0) / a);
		} else {
			tmp_4 = t_2;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	} else {
		tmp_1 = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = sqrt((a * (c * (-4.0d0))))
    t_1 = (-0.5d0) * ((b + b) / a)
    t_2 = -c / b
    if (b <= (-2.2d-43)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = c * (2.0d0 / ((((a * 2.0d0) / (b / c)) - b) - b))
        end if
        tmp_1 = tmp_2
    else if (b <= (-5d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = c * (2.0d0 / (t_0 - b))
        end if
        tmp_1 = tmp_3
    else if (b <= 1d-8) then
        if (b >= 0.0d0) then
            tmp_4 = (-0.5d0) * ((b + t_0) / a)
        else
            tmp_4 = t_2
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (-0.5d0) * (((-2.0d0) * (c / b)) + (2.0d0 * (b / a)))
    else
        tmp_1 = t_2
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * (c * -4.0)));
	double t_1 = -0.5 * ((b + b) / a);
	double t_2 = -c / b;
	double tmp_1;
	if (b <= -2.2e-43) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = c * (2.0 / ((((a * 2.0) / (b / c)) - b) - b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = c * (2.0 / (t_0 - b));
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e-8) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * ((b + t_0) / a);
		} else {
			tmp_4 = t_2;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * (c * -4.0)))
	t_1 = -0.5 * ((b + b) / a)
	t_2 = -c / b
	tmp_1 = 0
	if b <= -2.2e-43:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = c * (2.0 / ((((a * 2.0) / (b / c)) - b) - b))
		tmp_1 = tmp_2
	elif b <= -5e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = c * (2.0 / (t_0 - b))
		tmp_1 = tmp_3
	elif b <= 1e-8:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = -0.5 * ((b + t_0) / a)
		else:
			tmp_4 = t_2
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)))
	else:
		tmp_1 = t_2
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(c * -4.0)))
	t_1 = Float64(-0.5 * Float64(Float64(b + b) / a))
	t_2 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -2.2e-43)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(Float64(Float64(a * 2.0) / Float64(b / c)) - b) - b)));
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(c * Float64(2.0 / Float64(t_0 - b)));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1e-8)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * Float64(Float64(b + t_0) / a));
		else
			tmp_4 = t_2;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(-0.5 * Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / a))));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = sqrt((a * (c * -4.0)));
	t_1 = -0.5 * ((b + b) / a);
	t_2 = -c / b;
	tmp_2 = 0.0;
	if (b <= -2.2e-43)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = c * (2.0 / ((((a * 2.0) / (b / c)) - b) - b));
		end
		tmp_2 = tmp_3;
	elseif (b <= -5e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = c * (2.0 / (t_0 - b));
		end
		tmp_2 = tmp_4;
	elseif (b <= 1e-8)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = -0.5 * ((b + t_0) / a);
		else
			tmp_5 = t_2;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	else
		tmp_2 = t_2;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -2.2e-43], If[GreaterEqual[b, 0.0], t$95$1, N[(c * N[(2.0 / N[(N[(N[(N[(a * 2.0), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(c * N[(2.0 / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e-8], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$2], 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], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.19999999999999997e-43

   1. Initial program 66.9%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in b around inf 66.8%

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

   5. Taylor expanded in b around -inf 83.8%

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

      1. +-commutative 83.8%

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

      2. mul-1-neg 83.8%

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

      3. unsub-neg 83.8%

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

      4. associate-/l* 88.9%

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

      5. associate-*r/ 88.9%

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

   7. Simplified 88.9%

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

   if -2.19999999999999997e-43 < b < -4.999999999999985e-310

   1. Initial program 78.7%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in b around inf 78.3%

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

   5. Taylor expanded in b around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*r* 69.7%

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

   7. Simplified 69.7%

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

   if -4.999999999999985e-310 < b < 1e-8

   1. Initial program 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in b around -inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. distribute-neg-frac 88.5%

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

   6. Simplified 88.5%

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

   7. Taylor expanded in b around 0 71.2%

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

      1. *-commutative 24.3%

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

      2. associate-*r* 24.3%

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

   9. Simplified 71.2%

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

   if 1e-8 < b

   1. Initial program 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in b around -inf 62.9%

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

      1. mul-1-neg 62.9%

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

      2. distribute-neg-frac 62.9%

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

   6. Simplified 62.9%

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

   7. Taylor expanded in b around inf 94.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(\frac{a \cdot 2}{\frac{b}{c}} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-8}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{b + b}{a}\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{-53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(\frac{a \cdot 2}{\frac{b}{c}} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (+ b b) a))))
   (if (<= b -4.2e-53)
     (if (>= b 0.0) t_0 (* c (/ 2.0 (- (- (/ (* a 2.0) (/ b c)) b) b))))
     (if (>= b 0.0) t_0 (* c (/ 2.0 (- (sqrt (* a (* c -4.0))) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = -0.5 * ((b + b) / a);
	double tmp_1;
	if (b <= -4.2e-53) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = c * (2.0 / ((((a * 2.0) / (b / c)) - b) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = c * (2.0 / (sqrt((a * (c * -4.0))) - b));
	}
	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 = (-0.5d0) * ((b + b) / a)
    if (b <= (-4.2d-53)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = c * (2.0d0 / ((((a * 2.0d0) / (b / c)) - b) - b))
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = c * (2.0d0 / (sqrt((a * (c * (-4.0d0)))) - b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -0.5 * ((b + b) / a);
	double tmp_1;
	if (b <= -4.2e-53) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = c * (2.0 / ((((a * 2.0) / (b / c)) - b) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = c * (2.0 / (Math.sqrt((a * (c * -4.0))) - b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -0.5 * ((b + b) / a)
	tmp_1 = 0
	if b <= -4.2e-53:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = c * (2.0 / ((((a * 2.0) / (b / c)) - b) - b))
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = c * (2.0 / (math.sqrt((a * (c * -4.0))) - b))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(-0.5 * Float64(Float64(b + b) / a))
	tmp_1 = 0.0
	if (b <= -4.2e-53)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(Float64(Float64(a * 2.0) / Float64(b / c)) - b) - b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(c * Float64(2.0 / Float64(sqrt(Float64(a * Float64(c * -4.0))) - b)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = -0.5 * ((b + b) / a);
	tmp_2 = 0.0;
	if (b <= -4.2e-53)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = c * (2.0 / ((((a * 2.0) / (b / c)) - b) - b));
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = c * (2.0 / (sqrt((a * (c * -4.0))) - b));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.2e-53], If[GreaterEqual[b, 0.0], t$95$0, N[(c * N[(2.0 / N[(N[(N[(N[(a * 2.0), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(c * N[(2.0 / N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.19999999999999955e-53

   1. Initial program 66.9%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in b around inf 66.8%

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

   5. Taylor expanded in b around -inf 83.8%

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

      1. +-commutative 83.8%

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

      2. mul-1-neg 83.8%

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

      3. unsub-neg 83.8%

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

      4. associate-/l* 88.9%

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

      5. associate-*r/ 88.9%

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

   7. Simplified 88.9%

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

   if -4.19999999999999955e-53 < b

   1. Initial program 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in b around inf 72.1%

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

   5. Taylor expanded in b around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. associate-*r* 70.7%

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

   7. Simplified 70.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(\frac{a \cdot 2}{\frac{b}{c}} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\\ \end{array} \]

Alternative 4: 67.6% accurate, 7.8× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -0.5 (+ (* -2.0 (/ c b)) (* 2.0 (/ b a)))) (/ (- c) 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 / 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 / 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 / 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 / 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) / 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 / 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[((-c) / b), $MachinePrecision]]

Derivation

1. Initial program 70.7%

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

3. Simplified 70.6%

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

4. Taylor expanded in b around -inf 70.7%

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

   1. mul-1-neg 70.7%

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

   2. distribute-neg-frac 70.7%

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

6. Simplified 70.7%

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

7. Taylor expanded in b around inf 71.3%

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

8. Final simplification 71.3%

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

Alternative 5: 67.4% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Simplified 70.6%

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

4. Taylor expanded in b around -inf 70.7%

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

   1. mul-1-neg 70.7%

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

   2. distribute-neg-frac 70.7%

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

6. Simplified 70.7%

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

7. Taylor expanded in b around inf 71.3%

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

8. Taylor expanded in c around 0 70.6%

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

   1. associate-*r/ 70.6%

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

   2. associate-/l* 70.4%

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

   3. associate-/r/ 70.5%

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

10. Simplified 70.5%

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

11. Taylor expanded in a around 0 70.6%

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

    1. associate-*r/ 70.6%

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

    2. mul-1-neg 70.6%

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

13. Simplified 70.6%

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

14. Final simplification 70.6%

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

Reproduce

herbie shell --seed 2023336 
(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)))))))