jeff quadratic root 2

Percentage Accurate: 71.6% → 90.2%

Time: 16.8s

Alternatives: 10

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "-9.229514494576979736667404585848881199935e-151")
  2. reg1 = (bf #e-4.204964517011546201293628214888038972742e112)
  3. reg2 = (bf "-3.588809910643040298190259878554032425954e198")

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: 71.6% accurate, 1.0× speedup

Math

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

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: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - t\_0}\\ t_2 := \frac{t\_0 - b}{2 \cdot a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_1 (/ (* 2.0 c) (- (- b) t_0)))
        (t_2 (/ (- t_0 b) (* 2.0 a))))
   (if (<= b -1e+128)
     (if (>= b 0.0) t_1 (/ (- (- b) b) (* 2.0 a)))
     (if (<= b 1.15e+64)
       (if (>= b 0.0) t_1 t_2)
       (if (>= b 0.0) (/ (* 2.0 c) (* b -2.0)) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (2.0 * c) / (-b - t_0);
	double t_2 = (t_0 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -1e+128) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (-b - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.15e+64) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = t_2;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * -2.0);
	} 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
    t_0 = sqrt(((b * b) - (c * (4.0d0 * a))))
    t_1 = (2.0d0 * c) / (-b - t_0)
    t_2 = (t_0 - b) / (2.0d0 * a)
    if (b <= (-1d+128)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = (-b - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 1.15d+64) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = t_2
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / (b * (-2.0d0))
    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(((b * b) - (c * (4.0 * a))));
	double t_1 = (2.0 * c) / (-b - t_0);
	double t_2 = (t_0 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -1e+128) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (-b - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.15e+64) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = t_2;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * -2.0);
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (4.0 * a))))
	t_1 = (2.0 * c) / (-b - t_0)
	t_2 = (t_0 - b) / (2.0 * a)
	tmp_1 = 0
	if b <= -1e+128:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = (-b - b) / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 1.15e+64:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = t_2
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (b * -2.0)
	else:
		tmp_1 = t_2
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0))
	t_2 = Float64(Float64(t_0 - b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -1e+128)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(Float64(-b) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.15e+64)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = t_2;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(b * -2.0));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	t_1 = (2.0 * c) / (-b - t_0);
	t_2 = (t_0 - b) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -1e+128)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = (-b - b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.15e+64)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = t_2;
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (b * -2.0);
	else
		tmp_2 = t_2;
	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[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e+128], If[GreaterEqual[b, 0.0], t$95$1, N[(N[((-b) - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.15e+64], If[GreaterEqual[b, 0.0], t$95$1, t$95$2], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0000000000000001e128

   1. Initial program 26.9%

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

      1. add-cbrt-cube 25.0%

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

      2. pow3 25.0%

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

      3. sqrt-pow2 25.0%

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

      4. fma-neg 25.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)}^{\left(\frac{3}{2}\right)}}}{2 \cdot a}\\ \end{array} \]

      5. *-commutative 25.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{2 \cdot a}\\ \end{array} \]

      6. distribute-rgt-neg-in 25.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{2 \cdot a}\\ \end{array} \]

      7. distribute-lft-neg-in 25.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(\left(-4\right) \cdot a\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{2 \cdot a}\\ \end{array} \]

      8. metadata-eval 25.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{2 \cdot a}\\ \end{array} \]

      9. *-commutative 25.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{2 \cdot a}\\ \end{array} \]

      10. metadata-eval 25.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}}}{2 \cdot a}\\ \end{array} \]

   4. Applied egg-rr 25.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in b around -inf 96.5%

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

   if -1.0000000000000001e128 < b < 1.15e64

   1. Initial program 85.2%

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

   if 1.15e64 < b

   1. Initial program 50.8%

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

   3. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \frac{t\_0 - b}{2 \cdot a}\\ \mathbf{if}\;b \leq -9 \cdot 10^{+127}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a))))) (t_1 (/ (- t_0 b) (* 2.0 a))))
   (if (<= b -9e+127)
     (if (>= b 0.0)
       (* c (* (/ -4.0 a) (/ b (* c -4.0))))
       (* 2.0 (* b (/ (cbrt -0.125) a))))
     (if (<= b -2e-310)
       (if (>= b 0.0) (/ b a) t_1)
       (if (<= b 1.1e+64)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) t_0))
           (* 2.0 (/ (* b (cbrt -0.125)) a)))
         (if (>= b 0.0) (/ (* 2.0 c) (* b -2.0)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (t_0 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -9e+127) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = 2.0 * (b * (cbrt(-0.125) / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.1e+64) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_4 = 2.0 * ((b * cbrt(-0.125)) / a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * -2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (t_0 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -9e+127) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = 2.0 * (b * (Math.cbrt(-0.125) / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.1e+64) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_4 = 2.0 * ((b * Math.cbrt(-0.125)) / a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * -2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_1 = Float64(Float64(t_0 - b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -9e+127)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(Float64(-4.0 / a) * Float64(b / Float64(c * -4.0))));
		else
			tmp_2 = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b / a);
		else
			tmp_3 = t_1;
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.1e+64)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_4 = Float64(2.0 * Float64(Float64(b * cbrt(-0.125)) / a));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(b * -2.0));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9e+127], If[GreaterEqual[b, 0.0], N[(c * N[(N[(-4.0 / a), $MachinePrecision] * N[(b / N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], t$95$1], If[LessEqual[b, 1.1e+64], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b * N[Power[-0.125, 1/3], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -9.00000000000000068e127

   1. Initial program 26.9%

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

      1. add-cbrt-cube 26.9%

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

      2. pow1/3 7.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 8.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 95.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-/l* 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. add-sqr-sqrt 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. sqrt-unprod 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      4. sqr-neg 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{b \cdot b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      5. sqrt-prod 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      6. add-sqr-sqrt 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{b} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      7. pow2 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{\color{blue}{{b}^{2}} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      8. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      9. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. Applied egg-rr 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. associate-*r* 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       9. associate-*r* 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       10. sub-neg 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{{b}^{2} + \left(-c \cdot \left(a \cdot 4\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       11. unpow2 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{b \cdot b} + \left(-c \cdot \left(a \cdot 4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       12. rem-square-sqrt 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{-c \cdot \left(a \cdot 4\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       13. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       14. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       15. metadata-eval 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       16. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   11. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   12. Taylor expanded in b around inf 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(-4 \cdot \frac{b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-4 \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{{\left(\sqrt{-4}\right)}^{2}} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. times-frac 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{{\left(\sqrt{-4}\right)}^{2}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{-4}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{-4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   14. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if -9.00000000000000068e127 < b < -1.999999999999994e-310

   1. Initial program 84.0%

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

   3. Taylor expanded in a around 0 84.0%

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

      1. distribute-lft-out-- 84.0%

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

      2. associate-/l* 84.0%

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

      3. fma-neg 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in c around inf 84.0%

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

   if -1.999999999999994e-310 < b < 1.10000000000000001e64

   1. Initial program 87.3%

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

      1. add-cbrt-cube 87.3%

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

      2. pow1/3 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 87.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 87.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]

   if 1.10000000000000001e64 < b

   1. Initial program 50.8%

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

   3. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+127}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_2 := \frac{t\_1 - b}{2 \cdot a}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 2.0 (* b (/ (cbrt -0.125) a))))
        (t_1 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_2 (/ (- t_1 b) (* 2.0 a))))
   (if (<= b -1.1e+126)
     (if (>= b 0.0) (* c (* (/ -4.0 a) (/ b (* c -4.0)))) t_0)
     (if (<= b -2e-310)
       (if (>= b 0.0) (/ b a) t_2)
       (if (<= b 1.15e+64)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) t_0)
         (if (>= b 0.0) (/ (* 2.0 c) (* b -2.0)) t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = 2.0 * (b * (cbrt(-0.125) / a));
	double t_1 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_2 = (t_1 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -1.1e+126) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = t_2;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.15e+64) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * -2.0);
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = 2.0 * (b * (Math.cbrt(-0.125) / a));
	double t_1 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double t_2 = (t_1 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -1.1e+126) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = t_2;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.15e+64) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * -2.0);
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_2 = Float64(Float64(t_1 - b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -1.1e+126)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(Float64(-4.0 / a) * Float64(b / Float64(c * -4.0))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b / a);
		else
			tmp_3 = t_2;
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.15e+64)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(b * -2.0));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e+126], If[GreaterEqual[b, 0.0], N[(c * N[(N[(-4.0 / a), $MachinePrecision] * N[(b / N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], t$95$2], If[LessEqual[b, 1.15e+64], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.09999999999999999e126

   1. Initial program 26.9%

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

      1. add-cbrt-cube 26.9%

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

      2. pow1/3 7.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 8.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 95.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-/l* 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. add-sqr-sqrt 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. sqrt-unprod 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      4. sqr-neg 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{b \cdot b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      5. sqrt-prod 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      6. add-sqr-sqrt 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{b} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      7. pow2 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{\color{blue}{{b}^{2}} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      8. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      9. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. Applied egg-rr 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. associate-*r* 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       9. associate-*r* 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       10. sub-neg 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{{b}^{2} + \left(-c \cdot \left(a \cdot 4\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       11. unpow2 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{b \cdot b} + \left(-c \cdot \left(a \cdot 4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       12. rem-square-sqrt 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{-c \cdot \left(a \cdot 4\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       13. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       14. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       15. metadata-eval 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       16. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   11. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   12. Taylor expanded in b around inf 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(-4 \cdot \frac{b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-4 \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{{\left(\sqrt{-4}\right)}^{2}} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. times-frac 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{{\left(\sqrt{-4}\right)}^{2}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{-4}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{-4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   14. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if -1.09999999999999999e126 < b < -1.999999999999994e-310

   1. Initial program 84.0%

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

   3. Taylor expanded in a around 0 84.0%

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

      1. distribute-lft-out-- 84.0%

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

      2. associate-/l* 84.0%

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

      3. fma-neg 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in c around inf 84.0%

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

   if -1.999999999999994e-310 < b < 1.15e64

   1. Initial program 87.3%

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

      1. add-cbrt-cube 87.3%

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

      2. pow1/3 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 87.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 87.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 87.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if 1.15e64 < b

   1. Initial program 50.8%

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

   3. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+126}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+127}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 2.0 (* b (/ (cbrt -0.125) a)))))
   (if (<= b -8.6e+127)
     (if (>= b 0.0) (* c (* (/ -4.0 a) (/ b (* c -4.0)))) t_0)
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (/ b a)
         (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a)))
       (if (<= b 4.6e-110)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (* c (* a -4.0))))) t_0)
         (if (>= b 0.0) (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b))) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = 2.0 * (b * (cbrt(-0.125) / a));
	double tmp_1;
	if (b <= -8.6e+127) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.6e-110) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = 2.0 * (b * (Math.cbrt(-0.125) / a));
	double tmp_1;
	if (b <= -8.6e+127) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.6e-110) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - Math.sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)))
	tmp_1 = 0.0
	if (b <= -8.6e+127)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(Float64(-4.0 / a) * Float64(b / Float64(c * -4.0))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b / a);
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4.6e-110)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.6e+127], If[GreaterEqual[b, 0.0], N[(c * N[(N[(-4.0 / a), $MachinePrecision] * N[(b / N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.6e-110], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.59999999999999968e127

   1. Initial program 26.9%

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

      1. add-cbrt-cube 26.9%

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

      2. pow1/3 7.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 8.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 95.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-/l* 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. add-sqr-sqrt 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. sqrt-unprod 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      4. sqr-neg 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{b \cdot b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      5. sqrt-prod 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      6. add-sqr-sqrt 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{b} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      7. pow2 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{\color{blue}{{b}^{2}} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      8. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      9. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. Applied egg-rr 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. associate-*r* 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       9. associate-*r* 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       10. sub-neg 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{{b}^{2} + \left(-c \cdot \left(a \cdot 4\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       11. unpow2 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{b \cdot b} + \left(-c \cdot \left(a \cdot 4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       12. rem-square-sqrt 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{-c \cdot \left(a \cdot 4\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       13. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       14. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       15. metadata-eval 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       16. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   11. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   12. Taylor expanded in b around inf 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(-4 \cdot \frac{b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-4 \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{{\left(\sqrt{-4}\right)}^{2}} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. times-frac 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{{\left(\sqrt{-4}\right)}^{2}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{-4}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{-4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   14. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if -8.59999999999999968e127 < b < -1.999999999999994e-310

   1. Initial program 84.0%

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

   3. Taylor expanded in a around 0 84.0%

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

      1. distribute-lft-out-- 84.0%

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

      2. associate-/l* 84.0%

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

      3. fma-neg 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in c around inf 84.0%

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

   if -1.999999999999994e-310 < b < 4.6000000000000003e-110

   1. Initial program 79.4%

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

      1. add-cbrt-cube 79.4%

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

      2. pow1/3 79.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 79.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   8. Taylor expanded in b around 0 75.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative 75.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. *-commutative 75.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. associate-*r* 75.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   10. Simplified 75.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if 4.6000000000000003e-110 < b

   1. Initial program 64.0%

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

      1. add-cbrt-cube 64.0%

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

      2. pow1/3 64.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 64.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 64.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 64.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 64.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   8. Taylor expanded in a around 0 73.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. distribute-lft-out-- 73.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. associate-/l* 80.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   10. Simplified 80.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+127}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ t_1 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{-27}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 2.0 (* b (/ (cbrt -0.125) a)))) (t_1 (sqrt (* c (* a -4.0)))))
   (if (<= b -3e-27)
     (if (>= b 0.0) (* c (* (/ -4.0 a) (/ b (* c -4.0)))) t_0)
     (if (<= b -2e-310)
       (if (>= b 0.0) (/ b a) (/ (- t_1 b) (* 2.0 a)))
       (if (<= b 4.5e-110)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) t_0)
         (if (>= b 0.0) (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b))) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = 2.0 * (b * (cbrt(-0.125) / a));
	double t_1 = sqrt((c * (a * -4.0)));
	double tmp_1;
	if (b <= -3e-27) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (t_1 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.5e-110) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = 2.0 * (b * (Math.cbrt(-0.125) / a));
	double t_1 = Math.sqrt((c * (a * -4.0)));
	double tmp_1;
	if (b <= -3e-27) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (t_1 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.5e-110) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)))
	t_1 = sqrt(Float64(c * Float64(a * -4.0)))
	tmp_1 = 0.0
	if (b <= -3e-27)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(Float64(-4.0 / a) * Float64(b / Float64(c * -4.0))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b / a);
		else
			tmp_3 = Float64(Float64(t_1 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4.5e-110)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e-27], If[GreaterEqual[b, 0.0], N[(c * N[(N[(-4.0 / a), $MachinePrecision] * N[(b / N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.5e-110], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.0000000000000001e-27

   1. Initial program 55.9%

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

      1. add-cbrt-cube 33.1%

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

      2. pow1/3 14.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 5.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 87.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 87.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-/l* 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. add-sqr-sqrt 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. sqrt-unprod 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      4. sqr-neg 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{b \cdot b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      5. sqrt-prod 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      6. add-sqr-sqrt 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{b} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      7. pow2 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{\color{blue}{{b}^{2}} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      8. *-commutative 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      9. *-commutative 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. Applied egg-rr 87.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. associate-*r/ 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. associate-*r* 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       9. associate-*r* 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       10. sub-neg 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{{b}^{2} + \left(-c \cdot \left(a \cdot 4\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       11. unpow2 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{b \cdot b} + \left(-c \cdot \left(a \cdot 4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       12. rem-square-sqrt 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{-c \cdot \left(a \cdot 4\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       13. distribute-rgt-neg-in 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       14. distribute-rgt-neg-in 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       15. metadata-eval 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       16. distribute-rgt-neg-in 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   11. Simplified 87.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   12. Taylor expanded in b around inf 87.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(-4 \cdot \frac{b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r/ 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-4 \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. rem-square-sqrt 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. unpow2 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{{\left(\sqrt{-4}\right)}^{2}} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. times-frac 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{{\left(\sqrt{-4}\right)}^{2}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. unpow2 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. rem-square-sqrt 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{-4}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. unpow2 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. rem-square-sqrt 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{-4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   14. Simplified 87.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if -3.0000000000000001e-27 < b < -1.999999999999994e-310

   1. Initial program 77.4%

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

   3. Taylor expanded in a around 0 77.4%

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

      1. distribute-lft-out-- 77.4%

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

      2. associate-/l* 77.4%

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

      3. fma-neg 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in c around inf 77.4%

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

   7. Taylor expanded in b around 0 67.8%

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

      1. *-commutative 17.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. *-commutative 17.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. associate-*r* 17.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. Simplified 68.2%

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

   if -1.999999999999994e-310 < b < 4.5000000000000001e-110

   1. Initial program 79.4%

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

      1. add-cbrt-cube 79.4%

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

      2. pow1/3 79.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 79.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   8. Taylor expanded in b around 0 75.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative 75.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. *-commutative 75.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. associate-*r* 75.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   10. Simplified 75.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if 4.5000000000000001e-110 < b

   1. Initial program 64.0%

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

      1. add-cbrt-cube 64.0%

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

      2. pow1/3 64.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 64.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 64.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 64.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 64.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   8. Taylor expanded in a around 0 73.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. distribute-lft-out-- 73.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. associate-/l* 80.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   10. Simplified 80.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-27}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \frac{t\_0 - b}{2 \cdot a}\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a))))) (t_1 (/ (- t_0 b) (* 2.0 a))))
   (if (<= b -1.05e+128)
     (if (>= b 0.0)
       (* c (* (/ -4.0 a) (/ b (* c -4.0))))
       (* 2.0 (* b (/ (cbrt -0.125) a))))
     (if (<= b 1.15e+64)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) t_1)
       (if (>= b 0.0) (/ (* 2.0 c) (* b -2.0)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (t_0 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -1.05e+128) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = 2.0 * (b * (cbrt(-0.125) / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.15e+64) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * -2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (t_0 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -1.05e+128) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = 2.0 * (b * (Math.cbrt(-0.125) / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.15e+64) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * -2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_1 = Float64(Float64(t_0 - b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -1.05e+128)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(Float64(-4.0 / a) * Float64(b / Float64(c * -4.0))));
		else
			tmp_2 = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.15e+64)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = t_1;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(b * -2.0));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.05e+128], If[GreaterEqual[b, 0.0], N[(c * N[(N[(-4.0 / a), $MachinePrecision] * N[(b / N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.15e+64], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e128

   1. Initial program 26.9%

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

      1. add-cbrt-cube 26.9%

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

      2. pow1/3 7.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 8.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 95.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-/l* 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. add-sqr-sqrt 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. sqrt-unprod 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      4. sqr-neg 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{b \cdot b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      5. sqrt-prod 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      6. add-sqr-sqrt 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{b} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      7. pow2 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{\color{blue}{{b}^{2}} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      8. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      9. *-commutative 95.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. Applied egg-rr 95.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. associate-*r* 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. *-commutative 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       9. associate-*r* 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       10. sub-neg 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{{b}^{2} + \left(-c \cdot \left(a \cdot 4\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       11. unpow2 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{b \cdot b} + \left(-c \cdot \left(a \cdot 4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       12. rem-square-sqrt 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{-c \cdot \left(a \cdot 4\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       13. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       14. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       15. metadata-eval 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       16. distribute-rgt-neg-in 95.5%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   11. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   12. Taylor expanded in b around inf 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(-4 \cdot \frac{b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r/ 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-4 \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{{\left(\sqrt{-4}\right)}^{2}} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. times-frac 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{{\left(\sqrt{-4}\right)}^{2}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{-4}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. unpow2 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. rem-square-sqrt 95.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{-4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   14. Simplified 95.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if -1.05e128 < b < 1.15e64

   1. Initial program 85.2%

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

   if 1.15e64 < b

   1. Initial program 50.8%

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

   3. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-26}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-260}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-26)
   (if (>= b 0.0)
     (* c (* (/ -4.0 a) (/ b (* c -4.0))))
     (* 2.0 (* b (/ (cbrt -0.125) a))))
   (if (<= b 5e-260)
     (if (>= b 0.0) (/ b a) (/ (- (sqrt (* c (* a -4.0))) b) (* 2.0 a)))
     (if (>= b 0.0)
       (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b)))
       (* 2.0 (/ (* b (cbrt -0.125)) a))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.25e-26) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = 2.0 * (b * (cbrt(-0.125) / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 5e-260) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (sqrt((c * (a * -4.0))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = 2.0 * ((b * cbrt(-0.125)) / a);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.25e-26) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * ((-4.0 / a) * (b / (c * -4.0)));
		} else {
			tmp_2 = 2.0 * (b * (Math.cbrt(-0.125) / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 5e-260) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (Math.sqrt((c * (a * -4.0))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = 2.0 * ((b * Math.cbrt(-0.125)) / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.25e-26)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(Float64(-4.0 / a) * Float64(b / Float64(c * -4.0))));
		else
			tmp_2 = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5e-260)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b / a);
		else
			tmp_3 = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp_1 = Float64(2.0 * Float64(Float64(b * cbrt(-0.125)) / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-26], If[GreaterEqual[b, 0.0], N[(c * N[(N[(-4.0 / a), $MachinePrecision] * N[(b / N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5e-260], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(b * N[Power[-0.125, 1/3], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25000000000000005e-26

   1. Initial program 55.9%

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

      1. add-cbrt-cube 33.1%

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

      2. pow1/3 14.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 5.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 87.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 87.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-/l* 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. add-sqr-sqrt 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. sqrt-unprod 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      4. sqr-neg 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{b \cdot b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      5. sqrt-prod 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      6. add-sqr-sqrt 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{b} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      7. pow2 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{\color{blue}{{b}^{2}} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      8. *-commutative 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      9. *-commutative 87.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. Applied egg-rr 87.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. associate-*r/ 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. associate-*r* 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. *-commutative 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       9. associate-*r* 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       10. sub-neg 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{{b}^{2} + \left(-c \cdot \left(a \cdot 4\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       11. unpow2 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{b \cdot b} + \left(-c \cdot \left(a \cdot 4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       12. rem-square-sqrt 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{-c \cdot \left(a \cdot 4\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       13. distribute-rgt-neg-in 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       14. distribute-rgt-neg-in 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       15. metadata-eval 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       16. distribute-rgt-neg-in 87.4%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   11. Simplified 87.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   12. Taylor expanded in b around inf 87.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(-4 \cdot \frac{b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   13. Step-by-step derivation

       1. associate-*r/ 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-4 \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       2. rem-square-sqrt 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       3. unpow2 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{{\left(\sqrt{-4}\right)}^{2}} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       4. times-frac 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{{\left(\sqrt{-4}\right)}^{2}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       5. unpow2 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       6. rem-square-sqrt 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{-4}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       7. unpow2 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

       8. rem-square-sqrt 87.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{-4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   14. Simplified 87.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   if -1.25000000000000005e-26 < b < 5.0000000000000003e-260

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0 69.1%

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

      1. distribute-lft-out-- 69.1%

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

      2. associate-/l* 69.0%

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

      3. fma-neg 69.0%

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

   5. Simplified 69.0%

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

   6. Taylor expanded in c around inf 69.1%

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

   7. Taylor expanded in b around 0 60.6%

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

      1. *-commutative 24.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. *-commutative 24.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      3. associate-*r* 24.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. Simplified 60.9%

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

   if 5.0000000000000003e-260 < b

   1. Initial program 66.0%

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

      1. add-cbrt-cube 66.0%

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

      2. pow1/3 66.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

   4. Applied egg-rr 66.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

   5. Taylor expanded in b around inf 66.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 66.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

   7. Simplified 66.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   8. Taylor expanded in a around 0 65.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. distribute-lft-out-- 65.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

      2. associate-/l* 72.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   10. Simplified 72.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   11. Taylor expanded in b around 0 72.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-26}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-260}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b)))
   (* 2.0 (* b (/ (cbrt -0.125) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = 2.0 * (b * (cbrt(-0.125) / a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = 2.0 * (b * (Math.cbrt(-0.125) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.7%

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

   1. add-cbrt-cube 48.7%

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

   2. pow1/3 38.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

4. Applied egg-rr 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

5. Taylor expanded in b around inf 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
6. Step-by-step derivation

   1. associate-/l* 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

7. Simplified 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

8. Taylor expanded in a around 0 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
9. Step-by-step derivation

   1. distribute-lft-out-- 63.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   2. associate-/l* 66.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

10. Simplified 66.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
11. Add Preprocessing

Alternative 9: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{b} \cdot \frac{c}{-2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* (/ 2.0 b) (/ c -2.0)) (* 2.0 (* b (/ (cbrt -0.125) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 / b) * (c / -2.0);
	} else {
		tmp = 2.0 * (b * (cbrt(-0.125) / a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 / b) * (c / -2.0);
	} else {
		tmp = 2.0 * (b * (Math.cbrt(-0.125) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 / b) * Float64(c / -2.0));
	else
		tmp = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(2.0 / b), $MachinePrecision] * N[(c / -2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.7%

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

   1. add-cbrt-cube 48.7%

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

   2. pow1/3 38.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

4. Applied egg-rr 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

5. Taylor expanded in b around inf 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
6. Step-by-step derivation

   1. associate-/l* 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

7. Simplified 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

8. Taylor expanded in b around inf 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
9. Step-by-step derivation

   1. *-commutative 65.1%

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

10. Simplified 65.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{b \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
11. Step-by-step derivation

    1. times-frac 65.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2}{b} \cdot \frac{c}{-2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

12. Applied egg-rr 65.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2}{b} \cdot \frac{c}{-2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
13. Add Preprocessing

Alternative 10: 35.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* c (* (/ -4.0 a) (/ b (* c -4.0))))
   (* 2.0 (* b (/ (cbrt -0.125) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c * ((-4.0 / a) * (b / (c * -4.0)));
	} else {
		tmp = 2.0 * (b * (cbrt(-0.125) / a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c * ((-4.0 / a) * (b / (c * -4.0)));
	} else {
		tmp = 2.0 * (b * (Math.cbrt(-0.125) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(c * Float64(Float64(-4.0 / a) * Float64(b / Float64(c * -4.0))));
	else
		tmp = Float64(2.0 * Float64(b * Float64(cbrt(-0.125) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(c * N[(N[(-4.0 / a), $MachinePrecision] * N[(b / N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b * N[(N[Power[-0.125, 1/3], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.7%

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

   1. add-cbrt-cube 48.7%

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

   2. pow1/3 38.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{0.3333333333333333}\\ \end{array} \]

4. Applied egg-rr 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{-2 \cdot a}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

5. Taylor expanded in b around inf 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{b \cdot \sqrt[3]{-0.125}}{a}\\ \end{array} \]
6. Step-by-step derivation

   1. associate-/l* 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)}\\ \end{array} \]

7. Simplified 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
8. Step-by-step derivation

   1. associate-/l* 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   2. add-sqr-sqrt 35.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   3. sqrt-unprod 45.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   4. sqr-neg 45.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{b \cdot b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   5. sqrt-prod 52.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   6. add-sqr-sqrt 51.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{b} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   7. pow2 51.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{\color{blue}{{b}^{2}} - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   8. *-commutative 51.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

   9. *-commutative 51.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

9. Applied egg-rr 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
10. Step-by-step derivation

    1. associate-*r/ 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    2. *-commutative 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c \cdot 2}}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    3. associate-*r/ 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    4. associate-*r* 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    5. *-commutative 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    6. *-commutative 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    7. *-commutative 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    8. *-commutative 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    9. associate-*r* 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    10. sub-neg 51.4%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{{b}^{2} + \left(-c \cdot \left(a \cdot 4\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    11. unpow2 51.4%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{\color{blue}{b \cdot b} + \left(-c \cdot \left(a \cdot 4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    12. rem-square-sqrt 49.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{-c \cdot \left(a \cdot 4\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    13. distribute-rgt-neg-in 49.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    14. distribute-rgt-neg-in 49.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    15. metadata-eval 49.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \cdot \sqrt{-c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    16. distribute-rgt-neg-in 49.6%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{\color{blue}{c \cdot \left(-a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

11. Simplified 45.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

12. Taylor expanded in b around inf 35.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(-4 \cdot \frac{b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
13. Step-by-step derivation

    1. associate-*r/ 35.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-4 \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    2. rem-square-sqrt 35.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    3. unpow2 35.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{{\left(\sqrt{-4}\right)}^{2}} \cdot b}{a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    4. times-frac 35.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{{\left(\sqrt{-4}\right)}^{2}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    5. unpow2 35.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    6. rem-square-sqrt 35.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{\color{blue}{-4}}{a} \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    7. unpow2 35.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

    8. rem-square-sqrt 37.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \left(\frac{-4}{a} \cdot \frac{b}{c \cdot \color{blue}{-4}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]

14. Simplified 37.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\left(\frac{-4}{a} \cdot \frac{b}{c \cdot -4}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \frac{\sqrt[3]{-0.125}}{a}\right)\\ \end{array} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(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))))