quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.1% → 85.4%

Time: 9.0s

Alternatives: 8

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: 52.1% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+145)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.8e-19)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+145) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.8e-19) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / 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 <= (-2d+145)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.8d-19) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / 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 <= -2e+145) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.8e-19) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e+145:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.8e-19:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+145)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.8e-19)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e+145)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.8e-19)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+145], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.8e-19], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2e145

   1. Initial program 43.7%

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

      1. +-commutative 43.7%

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

      2. unsub-neg 43.7%

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

   3. Simplified 43.7%

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

   4. Taylor expanded in b_2 around -inf 95.3%

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

      1. *-commutative 95.3%

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

   6. Simplified 95.3%

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

   if -2e145 < b_2 < 1.8000000000000001e-19

   1. Initial program 77.7%

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

      1. +-commutative 77.7%

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

      2. unsub-neg 77.7%

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

   3. Simplified 77.7%

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

   if 1.8000000000000001e-19 < b_2

   1. Initial program 13.0%

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

      1. +-commutative 13.0%

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

      2. unsub-neg 13.0%

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

   3. Simplified 13.0%

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

      1. *-un-lft-identity 13.0%

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

      2. *-un-lft-identity 13.0%

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

      3. prod-diff 13.0%

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

      4. *-commutative 13.0%

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

      5. *-un-lft-identity 13.0%

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

      6. fma-neg 13.0%

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

      7. *-un-lft-identity 13.0%

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

      8. sub-neg 13.0%

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

      9. add-sqr-sqrt 8.7%

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

      10. hypot-def 24.7%

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

      11. *-commutative 24.7%

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

      12. distribute-rgt-neg-in 24.7%

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

      13. *-commutative 24.7%

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

      14. *-un-lft-identity 24.7%

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

   5. Applied egg-rr 24.7%

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

      1. fma-udef 24.7%

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

      2. *-rgt-identity 24.7%

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

      3. associate-+r+ 24.7%

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

      4. +-commutative 24.7%

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

      5. unsub-neg 24.7%

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

   7. Simplified 24.7%

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

   8. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   9. 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 88.6%

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

      5. associate-*r* 88.6%

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

      6. metadata-eval 88.6%

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

   10. Simplified 88.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+145}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{-110}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.3e-110)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 2.05e-35) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.3e-110) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.05e-35) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / 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 <= (-2.3d-110)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 2.05d-35) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / 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 <= -2.3e-110) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 2.05e-35) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.3e-110:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 2.05e-35:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.3e-110)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 2.05e-35)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.3e-110)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 2.05e-35)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.3e-110], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.05e-35], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.3000000000000001e-110

   1. Initial program 70.5%

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

      1. +-commutative 70.5%

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

      2. unsub-neg 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in b_2 around -inf 82.5%

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

   if -2.3000000000000001e-110 < b_2 < 2.05000000000000013e-35

   1. Initial program 69.5%

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

      1. +-commutative 69.5%

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

      2. unsub-neg 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in b_2 around 0 65.0%

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

      1. mul-1-neg 65.0%

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

      2. distribute-rgt-neg-out 65.0%

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

   6. Simplified 65.0%

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

   if 2.05000000000000013e-35 < b_2

   1. Initial program 13.9%

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

      1. +-commutative 13.9%

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

      2. unsub-neg 13.9%

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

   3. Simplified 13.9%

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

      1. *-un-lft-identity 13.9%

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

      2. *-un-lft-identity 13.9%

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

      3. prod-diff 13.9%

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

      4. *-commutative 13.9%

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

      5. *-un-lft-identity 13.9%

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

      6. fma-neg 13.9%

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

      7. *-un-lft-identity 13.9%

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

      8. sub-neg 13.9%

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

      9. add-sqr-sqrt 9.7%

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

      10. hypot-def 25.3%

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

      11. *-commutative 25.3%

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

      12. distribute-rgt-neg-in 25.3%

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

      13. *-commutative 25.3%

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

      14. *-un-lft-identity 25.3%

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

   5. Applied egg-rr 25.3%

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

      1. fma-udef 25.3%

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

      2. *-rgt-identity 25.3%

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

      3. associate-+r+ 25.3%

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

      4. +-commutative 25.3%

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

      5. unsub-neg 25.3%

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

   7. Simplified 25.3%

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

   8. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   9. 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 87.8%

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

      5. associate-*r* 87.8%

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

      6. metadata-eval 87.8%

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

   10. Simplified 87.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{-110}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-110}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.55e-110)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.4e-18) (/ (sqrt (* a (- c))) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.55e-110) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.4e-18) {
		tmp = sqrt((a * -c)) / a;
	} else {
		tmp = (c * -0.5) / 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 <= (-2.55d-110)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.4d-18) then
        tmp = sqrt((a * -c)) / a
    else
        tmp = (c * (-0.5d0)) / 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 <= -2.55e-110) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.4e-18) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.55e-110:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.4e-18:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.55e-110)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.4e-18)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.55e-110)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.4e-18)
		tmp = sqrt((a * -c)) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.55e-110], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.4e-18], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.5500000000000001e-110

   1. Initial program 70.5%

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

      1. +-commutative 70.5%

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

      2. unsub-neg 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in b_2 around -inf 82.5%

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

   if -2.5500000000000001e-110 < b_2 < 1.40000000000000006e-18

   1. Initial program 69.1%

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

      1. +-commutative 69.1%

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

      2. unsub-neg 69.1%

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

   3. Simplified 69.1%

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

      1. prod-diff 68.7%

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

      2. *-commutative 68.7%

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

      3. fma-def 68.7%

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

      4. associate-+l+ 68.7%

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

      5. distribute-rgt-neg-in 68.7%

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

      6. fma-def 68.7%

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

      7. *-commutative 68.7%

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

      8. fma-udef 68.7%

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

      9. distribute-lft-neg-in 68.7%

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

      10. *-commutative 68.7%

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

      11. distribute-rgt-neg-in 68.7%

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

      12. fma-def 68.7%

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

   5. Applied egg-rr 68.7%

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

   6. Taylor expanded in b_2 around 0 62.6%

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

      1. associate-*l/ 62.7%

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

      2. *-lft-identity 62.7%

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

      3. distribute-rgt1-in 63.1%

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

      4. metadata-eval 63.1%

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

      5. *-commutative 63.1%

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

      6. mul-1-neg 63.1%

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

   8. Simplified 63.1%

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

   if 1.40000000000000006e-18 < b_2

   1. Initial program 13.0%

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

      1. +-commutative 13.0%

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

      2. unsub-neg 13.0%

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

   3. Simplified 13.0%

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

      1. *-un-lft-identity 13.0%

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

      2. *-un-lft-identity 13.0%

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

      3. prod-diff 13.0%

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

      4. *-commutative 13.0%

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

      5. *-un-lft-identity 13.0%

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

      6. fma-neg 13.0%

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

      7. *-un-lft-identity 13.0%

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

      8. sub-neg 13.0%

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

      9. add-sqr-sqrt 8.7%

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

      10. hypot-def 24.7%

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

      11. *-commutative 24.7%

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

      12. distribute-rgt-neg-in 24.7%

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

      13. *-commutative 24.7%

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

      14. *-un-lft-identity 24.7%

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

   5. Applied egg-rr 24.7%

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

      1. fma-udef 24.7%

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

      2. *-rgt-identity 24.7%

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

      3. associate-+r+ 24.7%

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

      4. +-commutative 24.7%

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

      5. unsub-neg 24.7%

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

   7. Simplified 24.7%

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

   8. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   9. 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 88.6%

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

      5. associate-*r* 88.6%

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

      6. metadata-eval 88.6%

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

   10. Simplified 88.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-110}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4: 67.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c * -0.5) / 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 <= (-1d-309)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else
        tmp = (c * (-0.5d0)) / 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 <= -1e-309) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 69.1%

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

      1. +-commutative 69.1%

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

      2. unsub-neg 69.1%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in b_2 around -inf 66.9%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 33.3%

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

      1. +-commutative 33.3%

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

      2. unsub-neg 33.3%

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

   3. Simplified 33.3%

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

      1. *-un-lft-identity 33.3%

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

      2. *-un-lft-identity 33.3%

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

      3. prod-diff 33.3%

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

      4. *-commutative 33.3%

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

      5. *-un-lft-identity 33.3%

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

      6. fma-neg 33.3%

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

      7. *-un-lft-identity 33.3%

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

      8. sub-neg 33.3%

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

      9. add-sqr-sqrt 30.5%

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

      10. hypot-def 40.9%

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

      11. *-commutative 40.9%

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

      12. distribute-rgt-neg-in 40.9%

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

      13. *-commutative 40.9%

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

      14. *-un-lft-identity 40.9%

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

   5. Applied egg-rr 40.9%

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

      1. fma-udef 40.9%

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

      2. *-rgt-identity 40.9%

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

      3. associate-+r+ 40.9%

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

      4. +-commutative 40.9%

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

      5. unsub-neg 40.9%

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

   7. Simplified 40.9%

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

   8. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   9. 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 65.3%

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

      5. associate-*r* 65.3%

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

      6. metadata-eval 65.3%

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

   10. Simplified 65.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 5: 47.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.2e-294) (/ (- b_2) a) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.2e-294) {
		tmp = -b_2 / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 1.2d-294) then
        tmp = -b_2 / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.2e-294) {
		tmp = -b_2 / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.2e-294:
		tmp = -b_2 / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.2e-294)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.2e-294)
		tmp = -b_2 / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.2e-294], N[((-b$95$2) / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.19999999999999999e-294

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. unsub-neg 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in b_2 around 0 38.4%

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

      1. mul-1-neg 38.4%

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

      2. distribute-rgt-neg-out 38.4%

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

   6. Simplified 38.4%

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

   7. Taylor expanded in c around 0 26.3%

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

      1. associate-*r/ 26.3%

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

      2. neg-mul-1 26.3%

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

   9. Simplified 26.3%

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

   if 1.19999999999999999e-294 < b_2

   1. Initial program 32.3%

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

      1. +-commutative 32.3%

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

      2. unsub-neg 32.3%

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

   3. Simplified 32.3%

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

   4. Taylor expanded in b_2 around inf 66.2%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6: 47.0% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.2e-294) (/ (- b_2) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.2e-294) {
		tmp = -b_2 / a;
	} else {
		tmp = (c * -0.5) / 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 <= 1.2d-294) then
        tmp = -b_2 / a
    else
        tmp = (c * (-0.5d0)) / 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 <= 1.2e-294) {
		tmp = -b_2 / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.2e-294:
		tmp = -b_2 / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.2e-294)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.2e-294)
		tmp = -b_2 / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.2e-294], N[((-b$95$2) / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.19999999999999999e-294

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. unsub-neg 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in b_2 around 0 38.4%

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

      1. mul-1-neg 38.4%

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

      2. distribute-rgt-neg-out 38.4%

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

   6. Simplified 38.4%

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

   7. Taylor expanded in c around 0 26.3%

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

      1. associate-*r/ 26.3%

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

      2. neg-mul-1 26.3%

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

   9. Simplified 26.3%

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

   if 1.19999999999999999e-294 < b_2

   1. Initial program 32.3%

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

      1. +-commutative 32.3%

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

      2. unsub-neg 32.3%

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

   3. Simplified 32.3%

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

      1. *-un-lft-identity 32.3%

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

      2. *-un-lft-identity 32.3%

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

      3. prod-diff 32.3%

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

      4. *-commutative 32.3%

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

      5. *-un-lft-identity 32.3%

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

      6. fma-neg 32.3%

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

      7. *-un-lft-identity 32.3%

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

      8. sub-neg 32.3%

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

      9. add-sqr-sqrt 29.4%

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

      10. hypot-def 40.0%

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

      11. *-commutative 40.0%

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

      12. distribute-rgt-neg-in 40.0%

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

      13. *-commutative 40.0%

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

      14. *-un-lft-identity 40.0%

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

   5. Applied egg-rr 40.0%

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

      1. fma-udef 40.0%

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

      2. *-rgt-identity 40.0%

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

      3. associate-+r+ 40.0%

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

      4. +-commutative 40.0%

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

      5. unsub-neg 40.0%

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

   7. Simplified 40.0%

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

   8. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   9. 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 66.2%

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

      5. associate-*r* 66.2%

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

      6. metadata-eval 66.2%

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

   10. Simplified 66.2%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 7: 66.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.2e-294) (/ (* b_2 -2.0) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.2e-294) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / 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 <= 1.2d-294) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / 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 <= 1.2e-294) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.2e-294:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.2e-294)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.2e-294)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.2e-294], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.19999999999999999e-294

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. unsub-neg 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in b_2 around -inf 65.4%

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

      1. *-commutative 65.4%

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

   6. Simplified 65.4%

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

   if 1.19999999999999999e-294 < b_2

   1. Initial program 32.3%

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

      1. +-commutative 32.3%

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

      2. unsub-neg 32.3%

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

   3. Simplified 32.3%

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

      1. *-un-lft-identity 32.3%

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

      2. *-un-lft-identity 32.3%

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

      3. prod-diff 32.3%

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

      4. *-commutative 32.3%

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

      5. *-un-lft-identity 32.3%

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

      6. fma-neg 32.3%

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

      7. *-un-lft-identity 32.3%

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

      8. sub-neg 32.3%

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

      9. add-sqr-sqrt 29.4%

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

      10. hypot-def 40.0%

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

      11. *-commutative 40.0%

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

      12. distribute-rgt-neg-in 40.0%

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

      13. *-commutative 40.0%

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

      14. *-un-lft-identity 40.0%

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

   5. Applied egg-rr 40.0%

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

      1. fma-udef 40.0%

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

      2. *-rgt-identity 40.0%

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

      3. associate-+r+ 40.0%

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

      4. +-commutative 40.0%

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

      5. unsub-neg 40.0%

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

   7. Simplified 40.0%

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

   8. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}} \]
   9. 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 66.2%

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

      5. associate-*r* 66.2%

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

      6. metadata-eval 66.2%

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

   10. Simplified 66.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 1.2 \cdot 10^{-294}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 8: 14.7% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

   1. +-commutative 50.5%

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

   2. unsub-neg 50.5%

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

3. Simplified 50.5%

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

4. Taylor expanded in b_2 around 0 32.4%

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

   1. mul-1-neg 32.4%

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

   2. distribute-rgt-neg-out 32.4%

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

6. Simplified 32.4%

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

7. Taylor expanded in c around 0 14.3%

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

   1. associate-*r/ 14.3%

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

   2. neg-mul-1 14.3%

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

9. Simplified 14.3%

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

10. Final simplification 14.3%

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

Reproduce

herbie shell --seed 2023264 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))