quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.5% → 85.6%

Time: 9.4s

Alternatives: 7

Speedup: 15.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 51.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7.2 \cdot 10^{+100}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e+100)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 4e-70)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e+100) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4e-70) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7.2d+100)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 4d-70) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e+100) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4e-70) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.2e+100:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 4e-70:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.2e+100)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 4e-70)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.2e+100)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 4e-70)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.2e+100], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4e-70], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.2e100

   1. Initial program 39.2%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 39.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 39.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 97.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if -7.2e100 < b_2 < 3.99999999999999998e-70

   1. Initial program 79.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 79.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 79.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if 3.99999999999999998e-70 < b_2

   1. Initial program 18.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 18.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 18.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 18.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 86.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.2 \cdot 10^{+100}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4.5 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 2.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.5e-38)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))
   (if (<= b_2 2.6e-70) (/ (- (sqrt (* a (- c))) b_2) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-38) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 2.6e-70) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.5d-38)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    else if (b_2 <= 2.6d-70) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-38) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 2.6e-70) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.5e-38:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	elif b_2 <= 2.6e-70:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.5e-38)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	elseif (b_2 <= 2.6e-70)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.5e-38)
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	elseif (b_2 <= 2.6e-70)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.5e-38], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.6e-70], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.50000000000000009e-38

   1. Initial program 56.2%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 56.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 56.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 56.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 92.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -4.50000000000000009e-38 < b_2 < 2.60000000000000002e-70

   1. Initial program 74.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 74.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 74.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 68.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 68.6%

         \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}} - b_2}{a} \]

      2. distribute-rgt-neg-out 68.6%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   6. Simplified 68.6%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   if 2.60000000000000002e-70 < b_2

   1. Initial program 18.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 18.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 18.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 18.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 86.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.5 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 2.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3.7 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 7.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.7e-38)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))
   (if (<= b_2 7.6e-71) (/ (sqrt (* a (- c))) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.7e-38) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 7.6e-71) {
		tmp = sqrt((a * -c)) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.7d-38)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    else if (b_2 <= 7.6d-71) then
        tmp = sqrt((a * -c)) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.7e-38) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else if (b_2 <= 7.6e-71) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.7e-38:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	elif b_2 <= 7.6e-71:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.7e-38)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	elseif (b_2 <= 7.6e-71)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.7e-38)
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	elseif (b_2 <= 7.6e-71)
		tmp = sqrt((a * -c)) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.7e-38], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.6e-71], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.7e-38

   1. Initial program 56.2%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 56.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 56.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 56.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 92.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -3.7e-38 < b_2 < 7.59999999999999984e-71

   1. Initial program 74.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 74.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 74.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. prod-diff 73.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}} - b_2}{a} \]

      2. *-commutative 73.7%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)} - b_2}{a} \]

      3. fma-def 73.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(b_2 \cdot b_2 + \left(-a \cdot c\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)} - b_2}{a} \]

      4. associate-+l+ 73.7%

         \[\leadsto \frac{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(\left(-a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}} - b_2}{a} \]

      5. distribute-rgt-neg-in 73.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \left(\color{blue}{a \cdot \left(-c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} - b_2}{a} \]

      6. fma-def 73.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \color{blue}{\mathsf{fma}\left(a, -c, \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}} - b_2}{a} \]

      7. *-commutative 73.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)} - b_2}{a} \]

      8. fma-udef 73.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \color{blue}{\left(-c\right) \cdot a + a \cdot c}\right)} - b_2}{a} \]

      9. distribute-lft-neg-in 73.7%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)} - b_2}{a} \]

      10. *-commutative 73.7%

          \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)} - b_2}{a} \]

      11. distribute-rgt-neg-in 73.7%

          \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)} - b_2}{a} \]

      12. fma-def 73.7%

          \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)} - b_2}{a} \]

   5. Applied egg-rr 73.7%

      \[\leadsto \frac{\sqrt{\color{blue}{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}} - b_2}{a} \]

   6. Taylor expanded in b_2 around 0 66.2%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \sqrt{c \cdot a + -2 \cdot \left(c \cdot a\right)}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.4%

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

      2. *-lft-identity 66.4%

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

      3. distribute-rgt1-in 66.7%

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

      4. metadata-eval 66.7%

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

      5. mul-1-neg 66.7%

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

      6. *-commutative 66.7%

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

      7. distribute-rgt-neg-out 66.7%

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

   8. Simplified 66.7%

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

   if 7.59999999999999984e-71 < b_2

   1. Initial program 18.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 18.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 18.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 18.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 86.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.7 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 7.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4: 67.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))
   (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 65.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 65.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 65.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 67.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 33.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 33.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 33.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 64.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5: 67.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* -2.0 (/ b_2 a)) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 65.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 65.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 65.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 67.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 33.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 33.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 33.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 64.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6: 34.9% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -2.0 (/ b_2 a)))

C

double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-2.0d0) * (b_2 / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Python

def code(a, b_2, c):
	return -2.0 * (b_2 / a)

Julia

function code(a, b_2, c)
	return Float64(-2.0 * Float64(b_2 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -2.0 * (b_2 / a);
end

Wolfram

code[a_, b$95$2_, c_] := N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 50.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. unsub-neg 50.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

3. Simplified 50.9%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

4. Taylor expanded in b_2 around -inf 38.1%

   \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

5. Final simplification 38.1%

   \[\leadsto -2 \cdot \frac{b_2}{a} \]

Alternative 7: 15.1% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{-b_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (- b_2) a))

C

double code(double a, double b_2, double c) {
	return -b_2 / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = -b_2 / a
end function

Java

public static double code(double a, double b_2, double c) {
	return -b_2 / a;
}

Python

def code(a, b_2, c):
	return -b_2 / a

Julia

function code(a, b_2, c)
	return Float64(Float64(-b_2) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -b_2 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[((-b$95$2) / a), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 50.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. unsub-neg 50.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

3. Simplified 50.9%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
4. Step-by-step derivation

   1. div-sub 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}} \]

   2. add-cbrt-cube 41.6%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} - \frac{b_2}{a} \]

   3. add-sqr-sqrt 41.6%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} - \frac{b_2}{a} \]

   4. cbrt-prod 48.9%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} - \frac{b_2}{a} \]

   5. *-un-lft-identity 48.9%

      \[\leadsto \frac{\sqrt[3]{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{1 \cdot a}} - \frac{b_2}{a} \]

   6. times-frac 48.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}{1} \cdot \frac{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}} - \frac{b_2}{a} \]

   7. fma-neg 48.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}{1}, \frac{\sqrt[3]{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}, -\frac{b_2}{a}\right)} \]

5. Applied egg-rr 39.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{1}, \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a}, -\frac{b_2}{a}\right)} \]
6. Step-by-step derivation

   1. fma-udef 39.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{1} \cdot \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a} + \left(-\frac{b_2}{a}\right)} \]

   2. unsub-neg 39.9%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{1} \cdot \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a} - \frac{b_2}{a}} \]

   3. /-rgt-identity 39.9%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}} \cdot \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a} - \frac{b_2}{a} \]

   4. distribute-rgt-neg-out 39.9%

      \[\leadsto \sqrt[3]{\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)} \cdot \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a} - \frac{b_2}{a} \]

   5. *-commutative 39.9%

      \[\leadsto \sqrt[3]{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)} \cdot \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a} - \frac{b_2}{a} \]

   6. fma-neg 39.9%

      \[\leadsto \sqrt[3]{\color{blue}{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a} - \frac{b_2}{a} \]

   7. *-commutative 39.9%

      \[\leadsto \sqrt[3]{b_2 \cdot b_2 - \color{blue}{c \cdot a}} \cdot \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a} - \frac{b_2}{a} \]

7. Simplified 39.9%

   \[\leadsto \color{blue}{\sqrt[3]{b_2 \cdot b_2 - c \cdot a} \cdot \frac{\sqrt[3]{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a} - \frac{b_2}{a}} \]

8. Taylor expanded in c around inf 15.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
9. Step-by-step derivation

   1. neg-mul-1 15.1%

      \[\leadsto \color{blue}{-\frac{b_2}{a}} \]

   2. distribute-neg-frac 15.1%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

10. Simplified 15.1%

    \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

11. Final simplification 15.1%

    \[\leadsto \frac{-b_2}{a} \]

Reproduce

herbie shell --seed 2023178 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))