quad2m (problem 3.2.1, negative)

Percentage Accurate: 57.9% → 90.9%

Time: 33.6s

Alternatives: 9

Speedup: 15.9×

• could not determine a ground truth

  1. a = -6.345523484422262e+49
  2. b_2 = 4.1334891586105655e+249
  3. c = 6.759880953206116e+186

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: 57.9% 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: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -7.5 \cdot 10^{-23}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq -9 \cdot 10^{-192}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 120000000:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -7.5e-23)
     (* -0.5 (/ c b_2))
     (if (<= b_2 -1.8e-160)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 -9e-192)
         (* c (/ -0.5 b_2))
         (if (<= b_2 120000000.0)
           (/ (- (- b_2) t_0) a)
           (/ (* b_2 -2.0) a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -7.5e-23) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -1.8e-160) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -9e-192) {
		tmp = c * (-0.5 / b_2);
	} else if (b_2 <= 120000000.0) {
		tmp = (-b_2 - t_0) / a;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-7.5d-23)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-1.8d-160)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= (-9d-192)) then
        tmp = c * ((-0.5d0) / b_2)
    else if (b_2 <= 120000000.0d0) then
        tmp = (-b_2 - t_0) / a
    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 t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -7.5e-23) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -1.8e-160) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -9e-192) {
		tmp = c * (-0.5 / b_2);
	} else if (b_2 <= 120000000.0) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	tmp = 0
	if b_2 <= -7.5e-23:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -1.8e-160:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= -9e-192:
		tmp = c * (-0.5 / b_2)
	elif b_2 <= 120000000.0:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -7.5e-23)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -1.8e-160)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= -9e-192)
		tmp = Float64(c * Float64(-0.5 / b_2));
	elseif (b_2 <= 120000000.0)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -7.5e-23)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -1.8e-160)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= -9e-192)
		tmp = c * (-0.5 / b_2);
	elseif (b_2 <= 120000000.0)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -7.5e-23], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -1.8e-160], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -9e-192], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 120000000.0], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -7.4999999999999998e-23

   1. Initial program 12.4%

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

   2. Taylor expanded in b_2 around -inf 98.9%

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

   if -7.4999999999999998e-23 < b_2 < -1.7999999999999999e-160

   1. Initial program 51.4%

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

      1. add-cube-cbrt 50.3%

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

      2. sqrt-prod 50.4%

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

      3. pow2 50.4%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

   3. Applied egg-rr 50.4%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{2}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
   4. Step-by-step derivation

      1. unpow2 50.4%

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

      2. rem-sqrt-square 50.4%

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

      3. rem-square-sqrt 50.2%

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

      4. unpow1/2 50.2%

         \[\leadsto \frac{\left(-b_2\right) - \left|\color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      5. unpow1/2 50.2%

         \[\leadsto \frac{\left(-b_2\right) - \left|{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5} \cdot \color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5}}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      6. fabs-sqr 50.2%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left({\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5} \cdot {\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5}\right)} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      7. pow-sqr 50.4%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{\left(2 \cdot 0.5\right)}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      8. metadata-eval 50.4%

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

      9. unpow1 50.4%

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

      10. *-commutative 50.4%

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

      11. *-commutative 50.4%

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

   5. Simplified 50.4%

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

      1. flip-- 50.4%

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

      2. frac-2neg 50.4%

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

   7. Applied egg-rr 51.5%

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

      1. neg-sub0 51.5%

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

      2. associate--r- 75.3%

         \[\leadsto \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + c \cdot a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

      3. +-inverses 75.3%

         \[\leadsto \frac{\frac{0 - \left(\color{blue}{0} + c \cdot a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

      4. associate--r+ 75.3%

         \[\leadsto \frac{\frac{\color{blue}{\left(0 - 0\right) - c \cdot a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

      5. metadata-eval 75.3%

         \[\leadsto \frac{\frac{\color{blue}{0} - c \cdot a}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

      6. neg-sub0 75.3%

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

      7. distribute-rgt-neg-in 75.3%

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

   9. Simplified 75.3%

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

   if -1.7999999999999999e-160 < b_2 < -9.00000000000000048e-192

   1. Initial program 32.8%

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

      1. add-cube-cbrt 32.8%

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

      2. sqrt-prod 33.0%

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

      3. pow2 33.0%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

   3. Applied egg-rr 33.0%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{2}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
   4. Step-by-step derivation

      1. unpow2 33.0%

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

      2. rem-sqrt-square 33.0%

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

      3. rem-square-sqrt 32.7%

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

      4. unpow1/2 32.7%

         \[\leadsto \frac{\left(-b_2\right) - \left|\color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      5. unpow1/2 32.7%

         \[\leadsto \frac{\left(-b_2\right) - \left|{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5} \cdot \color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5}}\right| \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      6. fabs-sqr 32.7%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\left({\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5} \cdot {\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{0.5}\right)} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      7. pow-sqr 33.0%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt[3]{b_2 \cdot b_2 - a \cdot c}\right)}^{\left(2 \cdot 0.5\right)}} \cdot \sqrt{\sqrt[3]{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      8. metadata-eval 33.0%

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

      9. unpow1 33.0%

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

      10. *-commutative 33.0%

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

      11. *-commutative 33.0%

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

   5. Simplified 33.0%

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

   6. Taylor expanded in b_2 around -inf 63.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   7. Step-by-step derivation

      1. metadata-eval 63.1%

         \[\leadsto \color{blue}{\frac{-0.5}{1}} \cdot \frac{c}{b_2} \]

      2. times-frac 63.1%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{1 \cdot b_2}} \]

      3. *-commutative 63.1%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{1 \cdot b_2} \]

      4. times-frac 63.1%

         \[\leadsto \color{blue}{\frac{c}{1} \cdot \frac{-0.5}{b_2}} \]

      5. /-rgt-identity 63.1%

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

   8. Simplified 63.1%

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

   if -9.00000000000000048e-192 < b_2 < 1.2e8

   1. Initial program 90.9%

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

   if 1.2e8 < b_2

   1. Initial program 85.6%

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

   2. Taylor expanded in b_2 around inf 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 5 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.5 \cdot 10^{-23}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq -9 \cdot 10^{-192}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \mathbf{elif}\;b_2 \leq 120000000:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-192}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 120000000:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-192)
   (* -0.5 (/ c b_2))
   (if (<= b_2 120000000.0)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-192) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 120000000.0) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} 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 <= (-9d-192)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 120000000.0d0) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    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 <= -9e-192) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 120000000.0) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-192:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 120000000.0:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-192)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 120000000.0)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	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 <= -9e-192)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 120000000.0)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-192], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 120000000.0], 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[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.00000000000000048e-192

   1. Initial program 22.3%

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

   2. Taylor expanded in b_2 around -inf 86.8%

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

   if -9.00000000000000048e-192 < b_2 < 1.2e8

   1. Initial program 90.9%

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

   if 1.2e8 < b_2

   1. Initial program 85.6%

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

   2. Taylor expanded in b_2 around inf 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-192}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 120000000:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-192}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-192)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6.3e-108)
     (/ (- b_2 (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-192) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.3e-108) {
		tmp = (b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 <= (-9d-192)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 6.3d-108) then
        tmp = (b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-192) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.3e-108) {
		tmp = (b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-192:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 6.3e-108:
		tmp = (b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-192)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6.3e-108)
		tmp = 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(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e-192)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 6.3e-108)
		tmp = (b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-192], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6.3e-108], 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[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.00000000000000048e-192

   1. Initial program 22.3%

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

   2. Taylor expanded in b_2 around -inf 86.8%

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

   if -9.00000000000000048e-192 < b_2 < 6.2999999999999997e-108

   1. Initial program 83.9%

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

      1. sub-neg 83.9%

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

      2. +-commutative 83.9%

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

      3. add-sqr-sqrt 83.5%

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

      4. distribute-rgt-neg-in 83.5%

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

      5. add-sqr-sqrt 24.1%

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

      6. sqrt-unprod 75.9%

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

      7. sqr-neg 75.9%

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

      8. sqrt-prod 51.4%

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

      9. add-sqr-sqrt 75.9%

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

      10. fma-def 75.9%

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

   3. Applied egg-rr 76.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}, -{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}, b_2\right)}}{a} \]
   4. Step-by-step derivation

      1. fma-udef 76.0%

         \[\leadsto \frac{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25} \cdot \left(-{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right) + b_2}}{a} \]

      2. +-commutative 76.0%

         \[\leadsto \frac{\color{blue}{b_2 + {\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25} \cdot \left(-{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}}{a} \]

      3. distribute-rgt-neg-out 76.0%

         \[\leadsto \frac{b_2 + \color{blue}{\left(-{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}}{a} \]

      4. pow-sqr 76.1%

         \[\leadsto \frac{b_2 + \left(-\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(2 \cdot 0.25\right)}}\right)}{a} \]

      5. metadata-eval 76.1%

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

      6. exp-to-pow 71.9%

         \[\leadsto \frac{b_2 + \left(-\color{blue}{e^{\log \left(b_2 \cdot b_2 - a \cdot c\right) \cdot 0.5}}\right)}{a} \]

      7. unsub-neg 71.9%

         \[\leadsto \frac{\color{blue}{b_2 - e^{\log \left(b_2 \cdot b_2 - a \cdot c\right) \cdot 0.5}}}{a} \]

      8. exp-to-pow 76.1%

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

      9. unpow1/2 76.1%

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

      10. *-commutative 76.1%

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

   5. Simplified 76.1%

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

   if 6.2999999999999997e-108 < b_2

   1. Initial program 90.7%

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

   2. Taylor expanded in b_2 around inf 88.7%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-192}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4: 77.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2 + 0.5 \cdot \frac{c \cdot a}{b_2}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (* -0.5 (/ c b_2))
   (/ (+ (* b_2 -2.0) (* 0.5 (/ (* c a) b_2))) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = ((b_2 * -2.0) + (0.5 * ((c * a) / 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 <= (-4d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((b_2 * (-2.0d0)) + (0.5d0 * ((c * a) / 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 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = ((b_2 * -2.0) + (0.5 * ((c * a) / b_2))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = ((b_2 * -2.0) + (0.5 * ((c * a) / b_2))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(Float64(b_2 * -2.0) + Float64(0.5 * Float64(Float64(c * a) / b_2))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = ((b_2 * -2.0) + (0.5 * ((c * a) / b_2))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] + N[(0.5 * N[(N[(c * a), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.2%

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

   2. Taylor expanded in b_2 around -inf 78.6%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 89.9%

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

   2. Taylor expanded in b_2 around inf 72.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2 + 0.5 \cdot \frac{c \cdot a}{b_2}}{a}\\ \end{array} \]

Alternative 5: 77.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (* -0.5 (/ c b_2))
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 <= (-4d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.2%

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

   2. Taylor expanded in b_2 around -inf 78.6%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 89.9%

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

   2. Taylor expanded in b_2 around inf 72.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 6: 58.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (* -0.5 (/ c b_2)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		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 <= (-4d-310)) 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 <= -4e-310) {
		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 <= -4e-310:
		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 <= -4e-310)
		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 <= -4e-310)
		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, -4e-310], 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 < -3.999999999999988e-310

   1. Initial program 28.2%

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

   2. Taylor expanded in b_2 around -inf 78.6%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 89.9%

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

      1. add-sqr-sqrt 89.6%

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

      2. pow2 89.6%

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

      3. pow1/2 89.6%

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

      4. sqrt-pow1 89.7%

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

      5. metadata-eval 89.7%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 89.7%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around inf 33.8%

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

      1. mul-1-neg 33.8%

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

   6. Simplified 33.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 7: 77.2% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310) (* -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 <= -4e-310) {
		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 <= (-4d-310)) 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 <= -4e-310) {
		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 <= -4e-310:
		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 <= -4e-310)
		tmp = Float64(-0.5 * Float64(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 <= -4e-310)
		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, -4e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 28.2%

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

   2. Taylor expanded in b_2 around -inf 78.6%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 89.9%

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

   2. Taylor expanded in b_2 around inf 72.3%

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

      1. *-commutative 72.3%

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

   4. Simplified 72.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 8: 28.9% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{-160}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.1e-160) (/ 0.0 a) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.1e-160) {
		tmp = 0.0 / a;
	} 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 <= (-2.1d-160)) then
        tmp = 0.0d0 / a
    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 <= -2.1e-160) {
		tmp = 0.0 / a;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.1e-160:
		tmp = 0.0 / a
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.1e-160)
		tmp = Float64(0.0 / a);
	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 <= -2.1e-160)
		tmp = 0.0 / a;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.1e-160], N[(0.0 / a), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.1e-160

   1. Initial program 21.6%

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

      1. add-sqr-sqrt 17.4%

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

      2. pow2 17.4%

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

      3. pow1/2 17.4%

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

      4. sqrt-pow1 17.5%

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

      5. metadata-eval 17.5%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 17.5%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around -inf 20.4%

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

      1. distribute-lft1-in 20.4%

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

      2. metadata-eval 20.4%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 20.4%

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

   6. Simplified 20.4%

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

   if -2.1e-160 < b_2

   1. Initial program 86.0%

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

      1. add-sqr-sqrt 85.7%

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

      2. pow2 85.7%

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

      3. pow1/2 85.7%

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

      4. sqrt-pow1 85.7%

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

      5. metadata-eval 85.7%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 85.7%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around inf 29.5%

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

      1. mul-1-neg 29.5%

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

   6. Simplified 29.5%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{-160}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 9: 10.2% accurate, 37.3× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b_2, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 59.3%

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

   1. add-sqr-sqrt 57.4%

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

   2. pow2 57.4%

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

   3. pow1/2 57.4%

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

   4. sqrt-pow1 57.5%

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

   5. metadata-eval 57.5%

      \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

3. Applied egg-rr 57.5%

   \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

4. Taylor expanded in b_2 around -inf 9.8%

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

   1. distribute-lft1-in 9.8%

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

   2. metadata-eval 9.8%

      \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

   3. mul0-lft 9.8%

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

6. Simplified 9.8%

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

7. Final simplification 9.8%

   \[\leadsto \frac{0}{a} \]

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))