The quadratic formula (r2)

Percentage Accurate: 52.3% → 84.8%

Time: 12.3s

Alternatives: 8

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c \cdot \frac{1}{b \cdot b}, -\frac{b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2)
   (/ c (- b))
   (if (<= b 6.2e+98)
     (/ (+ b (sqrt (fma b b (* c (* a -4.0))))) (* a -2.0))
     (fma b (* c (/ 1.0 (* b b))) (- (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2) {
		tmp = c / -b;
	} else if (b <= 6.2e+98) {
		tmp = (b + sqrt(fma(b, b, (c * (a * -4.0))))) / (a * -2.0);
	} else {
		tmp = fma(b, (c * (1.0 / (b * b))), -(b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6.2e+98)
		tmp = Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))) / Float64(a * -2.0));
	else
		tmp = fma(b, Float64(c * Float64(1.0 / Float64(b * b))), Float64(-Float64(b / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6.2e+98], N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(b * N[(c * N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.20000000000000018

   1. Initial program 11.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 92.2

         \[\leadsto \frac{c}{\color{blue}{-b}} \]

   5. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

   if -5.20000000000000018 < b < 6.20000000000000038e98

   1. Initial program 74.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 74.2%

      \[\leadsto \color{blue}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}} \]

   if 6.20000000000000038e98 < b

   1. Initial program 60.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\frac{c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{b \cdot \frac{c}{{b}^{2}} + b \cdot \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

      3. distribute-neg-frac N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + b \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{a}} \]

      4. metadata-eval N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + b \cdot \frac{\color{blue}{-1}}{a} \]

      5. associate-/l* N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \color{blue}{\frac{b \cdot -1}{a}} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \frac{\color{blue}{-1 \cdot b}}{a} \]

      7. associate-*r/ N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{c}{b \cdot b}, \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{c}{b \cdot b}, \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)}\right) \]

      14. lower-/.f64 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

      \[\leadsto \mathsf{fma}\left(b, \frac{1}{b \cdot b} \cdot \color{blue}{c}, -\frac{b}{a}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c \cdot \frac{1}{b \cdot b}, -\frac{b}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+98}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c \cdot \frac{1}{b \cdot b}, -\frac{b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2)
   (/ c (- b))
   (if (<= b 6.2e+98)
     (* (+ b (sqrt (fma b b (* c (* a -4.0))))) (/ -0.5 a))
     (fma b (* c (/ 1.0 (* b b))) (- (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2) {
		tmp = c / -b;
	} else if (b <= 6.2e+98) {
		tmp = (b + sqrt(fma(b, b, (c * (a * -4.0))))) * (-0.5 / a);
	} else {
		tmp = fma(b, (c * (1.0 / (b * b))), -(b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6.2e+98)
		tmp = Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))) * Float64(-0.5 / a));
	else
		tmp = fma(b, Float64(c * Float64(1.0 / Float64(b * b))), Float64(-Float64(b / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6.2e+98], N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(b * N[(c * N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.20000000000000018

   1. Initial program 14.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 91.7

         \[\leadsto \frac{c}{\color{blue}{-b}} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{c}{-b}} \]

   if -5.20000000000000018 < b < 6.20000000000000038e98

   1. Initial program 75.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)} \]

   if 6.20000000000000038e98 < b

   1. Initial program 53.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. sub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(\frac{c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{b \cdot \frac{c}{{b}^{2}} + b \cdot \left(\mathsf{neg}\left(\frac{1}{a}\right)\right)} \]

      3. distribute-neg-frac N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + b \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{a}} \]

      4. metadata-eval N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + b \cdot \frac{\color{blue}{-1}}{a} \]

      5. associate-/l* N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \color{blue}{\frac{b \cdot -1}{a}} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \frac{\color{blue}{-1 \cdot b}}{a} \]

      7. associate-*r/ N/A

         \[\leadsto b \cdot \frac{c}{{b}^{2}} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{c}{b \cdot b}, \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)}\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{c}{b \cdot b}, \color{blue}{\mathsf{neg}\left(\frac{b}{a}\right)}\right) \]

      14. lower-/.f64 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

      \[\leadsto \mathsf{fma}\left(b, \frac{1}{b \cdot b} \cdot \color{blue}{c}, -\frac{b}{a}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+98}:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c \cdot \frac{1}{b \cdot b}, -\frac{b}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 70.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024228 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))