quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.0% → 85.4%

Time: 13.9s

Alternatives: 9

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: 52.0% 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: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-105)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5e+102)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -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 <= -3.2e-105) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e+102) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (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 <= (-3.2d-105)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5d+102) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (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 <= -3.2e-105) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e+102) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-105:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5e+102:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-105)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5e+102)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -3.2e-105)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5e+102)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-105], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e+102], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.19999999999999981e-105

   1. Initial program 17.2%

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

      1. frac-2neg 17.2%

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

      2. div-inv 17.1%

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

   3. Applied egg-rr 25.4%

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

   4. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 83.4%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 83.4%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 83.4%

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

   6. Simplified 83.4%

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

   if -3.19999999999999981e-105 < b_2 < 5e102

   1. Initial program 82.3%

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

   if 5e102 < b_2

   1. Initial program 52.5%

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

   2. Taylor expanded in b_2 around inf 98.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.6e-107)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.9e-72)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (+ (* -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 <= -1.6e-107) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.9e-72) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (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 <= (-1.6d-107)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.9d-72) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (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 <= -1.6e-107) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.9e-72) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.6e-107:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.9e-72:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.6e-107)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.9e-72)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -1.6e-107)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.9e-72)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.6e-107], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.9e-72], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.60000000000000006e-107

   1. Initial program 17.2%

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

      1. frac-2neg 17.2%

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

      2. div-inv 17.1%

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

   3. Applied egg-rr 25.4%

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

   4. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 83.4%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 83.4%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 83.4%

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

   6. Simplified 83.4%

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

   if -1.60000000000000006e-107 < b_2 < 2.89999999999999998e-72

   1. Initial program 78.7%

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

   2. Taylor expanded in b_2 around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. neg-mul-1 76.6%

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

      3. *-commutative 76.6%

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

   4. Simplified 76.6%

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

   if 2.89999999999999998e-72 < b_2

   1. Initial program 67.2%

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

   2. Taylor expanded in b_2 around inf 86.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-105}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-105)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.6e-74)
     (/ (- (sqrt (* a (- c)))) a)
     (+ (* -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 <= -1.15e-105) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.6e-74) {
		tmp = -sqrt((a * -c)) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (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 <= (-1.15d-105)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.6d-74) then
        tmp = -sqrt((a * -c)) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (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 <= -1.15e-105) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.6e-74) {
		tmp = -Math.sqrt((a * -c)) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e-105:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.6e-74:
		tmp = -math.sqrt((a * -c)) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-105)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.6e-74)
		tmp = Float64(Float64(-sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -1.15e-105)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.6e-74)
		tmp = -sqrt((a * -c)) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-105], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.6e-74], N[((-N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.15e-105

   1. Initial program 17.2%

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

      1. frac-2neg 17.2%

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

      2. div-inv 17.1%

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

   3. Applied egg-rr 25.4%

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

   4. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 83.4%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 83.4%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 83.4%

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

   6. Simplified 83.4%

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

   if -1.15e-105 < b_2 < 2.6000000000000001e-74

   1. Initial program 78.7%

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

      1. prod-diff 78.4%

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

      2. *-commutative 78.4%

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

      3. fma-neg 78.4%

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

      4. prod-diff 78.4%

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

      5. *-commutative 78.4%

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

      6. fma-neg 78.4%

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

      7. associate-+l+ 78.3%

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

      8. *-commutative 78.3%

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

      9. fma-udef 78.4%

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

      10. distribute-lft-neg-in 78.4%

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

      11. *-commutative 78.4%

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

      12. distribute-rgt-neg-in 78.4%

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

      13. fma-def 78.3%

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

      14. *-commutative 78.3%

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

      15. fma-udef 78.4%

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

      16. distribute-lft-neg-in 78.4%

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

      17. *-commutative 78.4%

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

      18. distribute-rgt-neg-in 78.4%

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

      19. fma-def 78.3%

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

   3. Applied egg-rr 78.3%

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

      1. count-2 78.3%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

   5. Simplified 78.3%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]

   6. Taylor expanded in b_2 around 0 75.6%

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

      1. mul-1-neg 75.6%

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

   8. Simplified 76.0%

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

   if 2.6000000000000001e-74 < b_2

   1. Initial program 67.2%

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

   2. Taylor expanded in b_2 around inf 86.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-105}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4: 68.3% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* -0.5 c) b_2)
   (+ (* -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 = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (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 = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (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 = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + (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(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 = (-0.5 * c) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + (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[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 28.1%

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

      1. frac-2neg 28.1%

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

      2. div-inv 28.1%

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

   3. Applied egg-rr 34.5%

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

   4. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 68.1%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 68.1%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 68.1%

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

   6. Simplified 68.1%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.7%

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

   2. Taylor expanded in b_2 around inf 67.8%

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

4. Final simplification 67.9%

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

Alternative 5: 48.3% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-306}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-306) (* -0.5 (/ c b_2)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-306) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / a;
	}
	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 <= (-2.8d-306)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-306) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-306:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-306)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-306)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-306], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.8000000000000001e-306

   1. Initial program 28.3%

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

   2. Taylor expanded in b_2 around -inf 68.6%

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

   if -2.8000000000000001e-306 < b_2

   1. Initial program 72.2%

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

   2. Taylor expanded in b_2 around 0 40.6%

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

      1. associate-*r* 40.6%

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

      2. neg-mul-1 40.6%

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

      3. *-commutative 40.6%

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

   4. Simplified 40.6%

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

   5. Taylor expanded in b_2 around inf 28.5%

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

      1. associate-*r/ 28.5%

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

      2. mul-1-neg 28.5%

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

   7. Simplified 28.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-306}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 6: 48.3% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-306}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-306) (/ (* -0.5 c) b_2) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-306) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -b_2 / a;
	}
	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 <= (-2.8d-306)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-306) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-306:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-306)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-306)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-306], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.8000000000000001e-306

   1. Initial program 28.3%

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

      1. frac-2neg 28.3%

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

      2. div-inv 28.3%

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

   3. Applied egg-rr 34.8%

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

   4. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 68.6%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 68.6%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 68.6%

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

   6. Simplified 68.6%

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

   if -2.8000000000000001e-306 < b_2

   1. Initial program 72.2%

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

   2. Taylor expanded in b_2 around 0 40.6%

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

      1. associate-*r* 40.6%

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

      2. neg-mul-1 40.6%

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

      3. *-commutative 40.6%

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

   4. Simplified 40.6%

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

   5. Taylor expanded in b_2 around inf 28.5%

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

      1. associate-*r/ 28.5%

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

      2. mul-1-neg 28.5%

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

   7. Simplified 28.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-306}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 7: 68.2% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-306}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-306) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-306) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	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 <= (-2.8d-306)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-306) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-306:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-306)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-306)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-306], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.8000000000000001e-306

   1. Initial program 28.3%

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

      1. frac-2neg 28.3%

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

      2. div-inv 28.3%

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

   3. Applied egg-rr 34.8%

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

   4. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} \]

      2. *-commutative 0.0%

         \[\leadsto \frac{0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} \]

      3. unpow2 0.0%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} \]

      4. rem-square-sqrt 68.6%

         \[\leadsto \frac{0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} \]

      5. associate-*r* 68.6%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot -1\right) \cdot c}}{b_2} \]

      6. metadata-eval 68.6%

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

   6. Simplified 68.6%

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

   if -2.8000000000000001e-306 < b_2

   1. Initial program 72.2%

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

   2. Taylor expanded in b_2 around inf 66.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   3. Step-by-step derivation

      1. *-commutative 66.8%

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

   4. Simplified 66.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-306}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8: 24.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-306) 0.0 (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-306) {
		tmp = 0.0;
	} else {
		tmp = -b_2 / a;
	}
	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 <= (-2.8d-306)) then
        tmp = 0.0d0
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-306) {
		tmp = 0.0;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-306:
		tmp = 0.0
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-306)
		tmp = 0.0;
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-306)
		tmp = 0.0;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-306], 0.0, N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.8000000000000001e-306

   1. Initial program 28.3%

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

      1. frac-2neg 28.3%

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

      2. div-inv 28.3%

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

   3. Applied egg-rr 34.8%

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

   4. Taylor expanded in b_2 around inf 2.5%

      \[\leadsto \color{blue}{\left(2 \cdot b_2\right)} \cdot \frac{1}{-a} \]
   5. Step-by-step derivation

      1. count-2 2.5%

         \[\leadsto \color{blue}{\left(b_2 + b_2\right)} \cdot \frac{1}{-a} \]

   6. Simplified 2.5%

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

      1. *-commutative 2.5%

         \[\leadsto \color{blue}{\frac{1}{-a} \cdot \left(b_2 + b_2\right)} \]

      2. flip-+ 0.0%

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

      3. frac-times 0.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{\left(-a\right) \cdot \left(b_2 - b_2\right)}} \]

      4. *-un-lft-identity 0.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 0.0%

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

      7. sqr-neg 0.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. add-cube-cbrt 5.1%

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

      11. fma-neg 6.3%

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

      12. add-sqr-sqrt 6.3%

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

      13. sqrt-unprod 6.3%

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

      14. sqr-neg 6.3%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 6.3%

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

      17. fma-def 6.3%

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

      18. add-cube-cbrt 6.3%

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

   8. Applied egg-rr 6.3%

      \[\leadsto \color{blue}{\frac{b_2 \cdot b_2 - b_2 \cdot b_2}{a \cdot \left(b_2 + b_2\right)}} \]
   9. Step-by-step derivation

      1. div-sub 5.2%

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

      2. +-inverses 20.9%

         \[\leadsto \color{blue}{0} \]

   10. Simplified 20.9%

       \[\leadsto \color{blue}{0} \]

   if -2.8000000000000001e-306 < b_2

   1. Initial program 72.2%

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

   2. Taylor expanded in b_2 around 0 40.6%

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

      1. associate-*r* 40.6%

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

      2. neg-mul-1 40.6%

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

      3. *-commutative 40.6%

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

   4. Simplified 40.6%

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

   5. Taylor expanded in b_2 around inf 28.5%

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

      1. associate-*r/ 28.5%

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

      2. mul-1-neg 28.5%

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

   7. Simplified 28.5%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-306}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 9: 11.7% accurate, 112.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 50.6%

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

   1. frac-2neg 50.6%

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

   2. div-inv 50.5%

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

3. Applied egg-rr 48.6%

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

4. Taylor expanded in b_2 around inf 35.0%

   \[\leadsto \color{blue}{\left(2 \cdot b_2\right)} \cdot \frac{1}{-a} \]
5. Step-by-step derivation

   1. count-2 35.0%

      \[\leadsto \color{blue}{\left(b_2 + b_2\right)} \cdot \frac{1}{-a} \]

6. Simplified 35.0%

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

   1. *-commutative 35.0%

      \[\leadsto \color{blue}{\frac{1}{-a} \cdot \left(b_2 + b_2\right)} \]

   2. flip-+ 0.0%

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

   3. frac-times 0.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{\left(-a\right) \cdot \left(b_2 - b_2\right)}} \]

   4. *-un-lft-identity 0.0%

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

   5. add-sqr-sqrt 0.0%

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

   6. sqrt-unprod 0.0%

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

   7. sqr-neg 0.0%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 0.0%

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

   10. add-cube-cbrt 3.2%

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

   11. fma-neg 3.9%

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

   12. add-sqr-sqrt 3.1%

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

   13. sqrt-unprod 3.9%

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

   14. sqr-neg 3.9%

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

   15. sqrt-unprod 0.8%

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

   16. add-sqr-sqrt 3.9%

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

   17. fma-def 3.9%

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

   18. add-cube-cbrt 3.9%

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

8. Applied egg-rr 3.9%

   \[\leadsto \color{blue}{\frac{b_2 \cdot b_2 - b_2 \cdot b_2}{a \cdot \left(b_2 + b_2\right)}} \]
9. Step-by-step derivation

   1. div-sub 3.3%

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

   2. +-inverses 11.5%

      \[\leadsto \color{blue}{0} \]

10. Simplified 11.5%

    \[\leadsto \color{blue}{0} \]

11. Final simplification 11.5%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))