quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.0% → 85.7%

Time: 11.6s

Alternatives: 10

Speedup: 15.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b_2 \cdot b_2 - c \cdot a\\ t_1 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -2.65 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{t_0 + 2 \cdot \left(a \cdot \left(c \cdot 0\right)\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \leq -5.1 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 1.15 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{t_0}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (- (* b_2 b_2) (* c a))) (t_1 (/ (* -0.5 c) b_2)))
   (if (<= b_2 -2.65e+87)
     t_1
     (if (<= b_2 -2.3e-64)
       (/ (/ (* c a) (- (sqrt (+ t_0 (* 2.0 (* a (* c 0.0))))) b_2)) a)
       (if (<= b_2 -5.1e-127)
         t_1
         (if (<= b_2 1.15e+97)
           (/ (- (- b_2) (sqrt t_0)) a)
           (/ (* b_2 -2.0) a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = (b_2 * b_2) - (c * a);
	double t_1 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -2.65e+87) {
		tmp = t_1;
	} else if (b_2 <= -2.3e-64) {
		tmp = ((c * a) / (sqrt((t_0 + (2.0 * (a * (c * 0.0))))) - b_2)) / a;
	} else if (b_2 <= -5.1e-127) {
		tmp = t_1;
	} else if (b_2 <= 1.15e+97) {
		tmp = (-b_2 - sqrt(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) :: t_1
    real(8) :: tmp
    t_0 = (b_2 * b_2) - (c * a)
    t_1 = ((-0.5d0) * c) / b_2
    if (b_2 <= (-2.65d+87)) then
        tmp = t_1
    else if (b_2 <= (-2.3d-64)) then
        tmp = ((c * a) / (sqrt((t_0 + (2.0d0 * (a * (c * 0.0d0))))) - b_2)) / a
    else if (b_2 <= (-5.1d-127)) then
        tmp = t_1
    else if (b_2 <= 1.15d+97) then
        tmp = (-b_2 - sqrt(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 = (b_2 * b_2) - (c * a);
	double t_1 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -2.65e+87) {
		tmp = t_1;
	} else if (b_2 <= -2.3e-64) {
		tmp = ((c * a) / (Math.sqrt((t_0 + (2.0 * (a * (c * 0.0))))) - b_2)) / a;
	} else if (b_2 <= -5.1e-127) {
		tmp = t_1;
	} else if (b_2 <= 1.15e+97) {
		tmp = (-b_2 - Math.sqrt(t_0)) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = (b_2 * b_2) - (c * a)
	t_1 = (-0.5 * c) / b_2
	tmp = 0
	if b_2 <= -2.65e+87:
		tmp = t_1
	elif b_2 <= -2.3e-64:
		tmp = ((c * a) / (math.sqrt((t_0 + (2.0 * (a * (c * 0.0))))) - b_2)) / a
	elif b_2 <= -5.1e-127:
		tmp = t_1
	elif b_2 <= 1.15e+97:
		tmp = (-b_2 - math.sqrt(t_0)) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(b_2 * b_2) - Float64(c * a))
	t_1 = Float64(Float64(-0.5 * c) / b_2)
	tmp = 0.0
	if (b_2 <= -2.65e+87)
		tmp = t_1;
	elseif (b_2 <= -2.3e-64)
		tmp = Float64(Float64(Float64(c * a) / Float64(sqrt(Float64(t_0 + Float64(2.0 * Float64(a * Float64(c * 0.0))))) - b_2)) / a);
	elseif (b_2 <= -5.1e-127)
		tmp = t_1;
	elseif (b_2 <= 1.15e+97)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(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 = (b_2 * b_2) - (c * a);
	t_1 = (-0.5 * c) / b_2;
	tmp = 0.0;
	if (b_2 <= -2.65e+87)
		tmp = t_1;
	elseif (b_2 <= -2.3e-64)
		tmp = ((c * a) / (sqrt((t_0 + (2.0 * (a * (c * 0.0))))) - b_2)) / a;
	elseif (b_2 <= -5.1e-127)
		tmp = t_1;
	elseif (b_2 <= 1.15e+97)
		tmp = (-b_2 - sqrt(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[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]}, If[LessEqual[b$95$2, -2.65e+87], t$95$1, If[LessEqual[b$95$2, -2.3e-64], N[(N[(N[(c * a), $MachinePrecision] / N[(N[Sqrt[N[(t$95$0 + N[(2.0 * N[(a * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -5.1e-127], t$95$1, If[LessEqual[b$95$2, 1.15e+97], N[(N[((-b$95$2) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.65000000000000002e87 or -2.3000000000000001e-64 < b_2 < -5.10000000000000019e-127

   1. Initial program 10.0%

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

   2. Taylor expanded in b_2 around -inf 88.2%

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

      1. associate-*r/ 88.2%

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

   4. Simplified 88.2%

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

   if -2.65000000000000002e87 < b_2 < -2.3000000000000001e-64

   1. Initial program 33.6%

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

      1. prod-diff 33.5%

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

      2. *-commutative 33.5%

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

      3. fma-neg 33.5%

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

      4. prod-diff 33.5%

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

      5. *-commutative 33.5%

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

      6. fma-neg 33.5%

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

      7. associate-+l+ 33.5%

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

      8. *-commutative 33.5%

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

      9. fma-udef 33.5%

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

      10. distribute-lft-neg-in 33.5%

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

      11. *-commutative 33.5%

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

      12. distribute-rgt-neg-in 33.5%

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

      13. fma-def 33.5%

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

      14. *-commutative 33.5%

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

      15. fma-udef 33.5%

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

      16. distribute-lft-neg-in 33.5%

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

      17. *-commutative 33.5%

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

      18. distribute-rgt-neg-in 33.5%

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

      19. fma-def 33.5%

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

   3. Applied egg-rr 33.5%

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

      1. *-commutative 33.5%

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

      2. count-2 33.5%

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

      3. *-commutative 33.5%

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

   5. Simplified 33.5%

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

      1. flip-- 33.5%

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

      2. add-sqr-sqrt 33.5%

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

      3. *-commutative 33.5%

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

      4. *-commutative 33.5%

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

      5. *-commutative 33.5%

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

      6. *-commutative 33.5%

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

   7. Applied egg-rr 33.5%

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

   9. Simplified 33.5%

      \[\leadsto \frac{\color{blue}{\frac{b_2 \cdot b_2 - \left(\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \left(a \cdot \left(0 \cdot c\right)\right)\right)}{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \left(a \cdot \left(0 \cdot c\right)\right)} + \left(-b_2\right)}}}{a} \]

   10. Taylor expanded in b_2 around 0 91.6%

       \[\leadsto \frac{\frac{\color{blue}{c \cdot a}}{\sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + 2 \cdot \left(a \cdot \left(0 \cdot c\right)\right)} + \left(-b_2\right)}}{a} \]

   if -5.10000000000000019e-127 < b_2 < 1.15000000000000003e97

   1. Initial program 83.2%

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

   if 1.15000000000000003e97 < b_2

   1. Initial program 43.4%

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

   2. Taylor expanded in b_2 around inf 93.7%

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

      1. *-commutative 93.7%

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

   4. Simplified 93.7%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \left(a \cdot \left(c \cdot 0\right)\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \leq -5.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.15 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e-126)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.1e+97)
     (/ (- (- 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 <= -1.2e-126) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.1e+97) {
		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 <= (-1.2d-126)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.1d+97) 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 <= -1.2e-126) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.1e+97) {
		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 <= -1.2e-126:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.1e+97:
		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 <= -1.2e-126)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.1e+97)
		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 <= -1.2e-126)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.1e+97)
		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, -1.2e-126], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.1e+97], 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 < -1.20000000000000003e-126

   1. Initial program 18.3%

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

   2. Taylor expanded in b_2 around -inf 84.0%

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

      1. associate-*r/ 84.0%

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

   4. Simplified 84.0%

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

   if -1.20000000000000003e-126 < b_2 < 1.1e97

   1. Initial program 83.2%

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

   if 1.1e97 < b_2

   1. Initial program 43.4%

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

   2. Taylor expanded in b_2 around inf 93.7%

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

      1. *-commutative 93.7%

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

   4. Simplified 93.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e-126)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.8e-93)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-126) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.8e-93) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.2d-126)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.8d-93) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-126) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.8e-93) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e-126:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.8e-93:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e-126)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.8e-93)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e-126)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.8e-93)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e-126], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.8e-93], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.20000000000000003e-126

   1. Initial program 18.3%

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

   2. Taylor expanded in b_2 around -inf 84.0%

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

      1. associate-*r/ 84.0%

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

   4. Simplified 84.0%

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

   if -1.20000000000000003e-126 < b_2 < 1.8000000000000001e-93

   1. Initial program 79.6%

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

   2. Taylor expanded in b_2 around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. distribute-rgt-neg-out 74.1%

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

   4. Simplified 74.1%

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

   if 1.8000000000000001e-93 < b_2

   1. Initial program 66.6%

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

   2. Taylor expanded in b_2 around inf 82.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3.9 \cdot 10^{-128}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.9e-128)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.4e-94)
     (/ (- (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.9e-128) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.4e-94) {
		tmp = -sqrt((c * -a)) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.9d-128)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.4d-94) then
        tmp = -sqrt((c * -a)) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.9e-128) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.4e-94) {
		tmp = -Math.sqrt((c * -a)) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.9e-128:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.4e-94:
		tmp = -math.sqrt((c * -a)) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.9e-128)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.4e-94)
		tmp = Float64(Float64(-sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.9e-128)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.4e-94)
		tmp = -sqrt((c * -a)) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.9e-128], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.4e-94], N[((-N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.89999999999999997e-128

   1. Initial program 18.3%

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

   2. Taylor expanded in b_2 around -inf 84.0%

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

      1. associate-*r/ 84.0%

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

   4. Simplified 84.0%

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

   if -3.89999999999999997e-128 < b_2 < 2.4e-94

   1. Initial program 79.6%

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

      1. prod-diff 79.3%

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

      2. *-commutative 79.3%

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

      3. fma-neg 79.3%

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

      4. prod-diff 79.3%

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

      5. *-commutative 79.3%

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

      6. fma-neg 79.3%

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

      7. associate-+l+ 79.2%

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

      8. *-commutative 79.2%

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

      9. fma-udef 79.3%

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

      10. distribute-lft-neg-in 79.3%

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

      11. *-commutative 79.3%

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

      12. distribute-rgt-neg-in 79.3%

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

      13. fma-def 79.2%

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

      14. *-commutative 79.2%

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

      15. fma-udef 79.3%

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

      16. distribute-lft-neg-in 79.3%

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

      17. *-commutative 79.3%

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

      18. distribute-rgt-neg-in 79.3%

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

      19. fma-def 79.2%

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

   3. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. count-2 79.2%

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

      3. *-commutative 79.2%

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

   5. Simplified 79.2%

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

   6. Taylor expanded in b_2 around 0 71.9%

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

      1. mul-1-neg 71.9%

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

      2. fma-neg 71.9%

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

      3. associate-*r* 71.9%

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

      4. neg-mul-1 71.9%

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

      5. *-commutative 71.9%

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

      6. *-commutative 71.9%

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

      7. fma-def 71.8%

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

      8. fma-neg 71.8%

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

      9. fma-def 71.9%

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

      10. *-commutative 71.9%

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

      11. neg-mul-1 71.9%

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

      12. associate-*r* 71.9%

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

      13. *-commutative 71.9%

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

      14. distribute-lft1-in 71.9%

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

      15. metadata-eval 71.9%

          \[\leadsto \frac{-\sqrt{2 \cdot \left(\color{blue}{0} \cdot \left(c \cdot a\right)\right) - c \cdot a}}{a} \]

      16. mul0-lft 72.2%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{0} - c \cdot a}}{a} \]

      17. metadata-eval 72.2%

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

      18. neg-sub0 72.2%

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

      19. distribute-lft-neg-in 72.2%

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

      20. *-commutative 72.2%

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

   8. Simplified 72.2%

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

   if 2.4e-94 < b_2

   1. Initial program 66.6%

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

   2. Taylor expanded in b_2 around inf 82.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.9 \cdot 10^{-128}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5: 68.5% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310)
   (/ (* -0.5 c) b_2)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 30.8%

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

   2. Taylor expanded in b_2 around -inf 68.9%

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

      1. associate-*r/ 68.9%

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

   4. Simplified 68.9%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 70.5%

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

   2. Taylor expanded in b_2 around inf 65.0%

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

4. Final simplification 67.0%

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

Alternative 6: 68.2% accurate, 15.9× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310) (* c (/ -0.5 b_2)) (/ -2.0 (/ a b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -2.0 / (a / 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-310)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = (-2.0d0) / (a / 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-310) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = -2.0 / (a / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(-2.0 / Float64(a / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = c * (-0.5 / b_2);
	else
		tmp = -2.0 / (a / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 30.8%

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

      1. frac-2neg 30.8%

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

      2. div-inv 30.8%

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

   3. Applied egg-rr 37.1%

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

   4. Taylor expanded in b_2 around -inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 68.9%

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

      5. associate-*r* 68.9%

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

      6. metadata-eval 68.9%

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

      7. associate-/l* 68.6%

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

   6. Simplified 68.6%

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

      1. associate-/r/ 68.7%

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

   8. Applied egg-rr 68.7%

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

   if -1.999999999999994e-310 < b_2

   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. prod-diff 70.1%

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

      2. *-commutative 70.1%

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

      3. fma-neg 70.1%

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

      4. prod-diff 70.1%

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

      5. *-commutative 70.1%

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

      6. fma-neg 70.1%

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

      7. associate-+l+ 70.1%

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

      8. *-commutative 70.1%

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

      9. fma-udef 70.1%

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

      10. distribute-lft-neg-in 70.1%

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

      11. *-commutative 70.1%

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

      12. distribute-rgt-neg-in 70.1%

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

      13. fma-def 70.1%

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

      14. *-commutative 70.1%

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

      15. fma-udef 70.1%

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

      16. distribute-lft-neg-in 70.1%

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

      17. *-commutative 70.1%

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

      18. distribute-rgt-neg-in 70.1%

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

      19. fma-def 70.1%

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

   3. Applied egg-rr 70.1%

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

      1. *-commutative 70.1%

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

      2. count-2 70.1%

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

      3. *-commutative 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in b_2 around inf 64.6%

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

      1. associate-*r/ 64.6%

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

      2. associate-/l* 64.4%

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

   8. Simplified 64.4%

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

4. Final simplification 66.6%

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

Alternative 7: 68.3% accurate, 15.9× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310) (/ (* -0.5 c) b_2) (/ -2.0 (/ a b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 / (a / 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-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (-2.0d0) / (a / 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-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -2.0 / (a / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(-2.0 / Float64(a / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -2.0 / (a / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 30.8%

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

   2. Taylor expanded in b_2 around -inf 68.9%

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

      1. associate-*r/ 68.9%

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

   4. Simplified 68.9%

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

   if -1.999999999999994e-310 < b_2

   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. prod-diff 70.1%

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

      2. *-commutative 70.1%

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

      3. fma-neg 70.1%

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

      4. prod-diff 70.1%

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

      5. *-commutative 70.1%

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

      6. fma-neg 70.1%

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

      7. associate-+l+ 70.1%

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

      8. *-commutative 70.1%

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

      9. fma-udef 70.1%

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

      10. distribute-lft-neg-in 70.1%

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

      11. *-commutative 70.1%

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

      12. distribute-rgt-neg-in 70.1%

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

      13. fma-def 70.1%

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

      14. *-commutative 70.1%

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

      15. fma-udef 70.1%

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

      16. distribute-lft-neg-in 70.1%

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

      17. *-commutative 70.1%

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

      18. distribute-rgt-neg-in 70.1%

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

      19. fma-def 70.1%

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

   3. Applied egg-rr 70.1%

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

      1. *-commutative 70.1%

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

      2. count-2 70.1%

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

      3. *-commutative 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in b_2 around inf 64.6%

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

      1. associate-*r/ 64.6%

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

      2. associate-/l* 64.4%

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

   8. Simplified 64.4%

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

4. Final simplification 66.7%

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

Alternative 8: 68.3% accurate, 15.9× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-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 <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-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, -2e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 30.8%

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

   2. Taylor expanded in b_2 around -inf 68.9%

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

      1. associate-*r/ 68.9%

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

   4. Simplified 68.9%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 70.5%

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

   2. Taylor expanded in b_2 around inf 64.6%

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

      1. *-commutative 64.6%

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

   4. Simplified 64.6%

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

4. Final simplification 66.8%

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

Alternative 9: 10.6% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{0.5}{b_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* c (/ 0.5 b_2)))

C

double code(double a, double b_2, double c) {
	return c * (0.5 / b_2);
}

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 = c * (0.5d0 / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return c * (0.5 / b_2);
}

Python

def code(a, b_2, c):
	return c * (0.5 / b_2)

Julia

function code(a, b_2, c)
	return Float64(c * Float64(0.5 / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = c * (0.5 / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

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

2. Taylor expanded in b_2 around inf 30.6%

   \[\leadsto \frac{\color{blue}{-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}}}{a} \]
3. Step-by-step derivation

   1. div-inv 30.6%

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

   2. *-commutative 30.6%

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

4. Applied egg-rr 30.6%

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

   1. associate-*l* 33.3%

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

6. Simplified 33.3%

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

7. Taylor expanded in b_2 around 0 9.5%

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

   1. *-commutative 9.5%

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

   2. associate-*l/ 9.5%

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

   3. associate-*r/ 9.5%

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

9. Simplified 9.5%

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

10. Final simplification 9.5%

    \[\leadsto c \cdot \frac{0.5}{b_2} \]

Alternative 10: 34.7% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* c (/ -0.5 b_2)))

C

double code(double a, double b_2, double c) {
	return c * (-0.5 / b_2);
}

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 = c * ((-0.5d0) / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return c * (-0.5 / b_2);
}

Python

def code(a, b_2, c):
	return c * (-0.5 / b_2)

Julia

function code(a, b_2, c)
	return Float64(c * Float64(-0.5 / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = c * (-0.5 / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $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. frac-2neg 50.5%

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

   2. div-inv 50.3%

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

3. Applied egg-rr 47.8%

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

4. Taylor expanded in b_2 around -inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 35.8%

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

   5. associate-*r* 35.8%

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

   6. metadata-eval 35.8%

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

   7. associate-/l* 35.7%

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

6. Simplified 35.7%

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

   1. associate-/r/ 35.8%

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

8. Applied egg-rr 35.8%

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

9. Final simplification 35.8%

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

Reproduce

herbie shell --seed 2023171 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))