Quadratic roots, narrow range

Percentage Accurate: 55.2% → 92.3%

Time: 12.3s

Alternatives: 12

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \left(--0.5\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b} - \frac{\left(-c\right) \cdot c}{{b}^{5}} \cdot a, \frac{0.5}{b}\right), \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.027)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      0.5
      (fma
       a
       (fma
        a
        (- (* (- -0.5) (/ c (* (* b b) b))) (* (/ (* (- c) c) (pow b 5.0)) a))
        (/ 0.5 b))
       (* (/ b c) -0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.027) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = 0.5 / fma(a, fma(a, ((-(-0.5) * (c / ((b * b) * b))) - (((-c * c) / pow(b, 5.0)) * a)), (0.5 / b)), ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.027)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(0.5 / fma(a, fma(a, Float64(Float64(Float64(-(-0.5)) * Float64(c / Float64(Float64(b * b) * b))) - Float64(Float64(Float64(Float64(-c) * c) / (b ^ 5.0)) * a)), Float64(0.5 / b)), Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.027], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(a * N[(a * N[(N[((--0.5) * N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[((-c) * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0269999999999999997

   1. Initial program 87.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      4. associate-/r* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\color{blue}{\frac{1}{2}}}{a} \]

      6. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{\frac{1}{2}}{a} \]

      7. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot \frac{\frac{1}{2}}{a} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \cdot \frac{\frac{1}{2}}{a} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{2}}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot a}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot a} \]

   4. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b} \cdot a}} \]

   6. Applied rewrites 88.7%

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

   if 0.0269999999999999997 < b

   1. Initial program 51.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      4. associate-/r* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\color{blue}{\frac{1}{2}}}{a} \]

      6. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{\frac{1}{2}}{a} \]

      7. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot \frac{\frac{1}{2}}{a} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \cdot \frac{\frac{1}{2}}{a} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{2}}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot a}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot a} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b} \cdot a}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{-1}{2} \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(-1 \cdot \frac{c \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-1}{8} \cdot \frac{b \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right)}} \]

   7. Applied rewrites 94.1%

      \[\leadsto \frac{0.5}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \left(-a \cdot \left(\mathsf{fma}\left(-0.125, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{c \cdot c}, \frac{c \cdot c}{{b}^{5}}\right) + \left(-c \cdot \frac{\frac{c}{\left(b \cdot b\right) \cdot b} \cdot -0.5}{b \cdot b}\right)\right)\right) - \frac{c}{\left(b \cdot b\right) \cdot b} \cdot -0.5, \frac{0.5}{b}\right), -0.5 \cdot \frac{b}{c}\right)}} \]

   8. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{2}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \left(\mathsf{neg}\left(a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \frac{c}{\left(b \cdot b\right) \cdot b} \cdot \frac{-1}{2}, \frac{\frac{1}{2}}{b}\right), \frac{-1}{2} \cdot \frac{b}{c}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.1%

       \[\leadsto \frac{0.5}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \left(-a \cdot \frac{-c \cdot c}{{b}^{5}}\right) - \frac{c}{\left(b \cdot b\right) \cdot b} \cdot -0.5, \frac{0.5}{b}\right), -0.5 \cdot \frac{b}{c}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \left(--0.5\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b} - \frac{\left(-c\right) \cdot c}{{b}^{5}} \cdot a, \frac{0.5}{b}\right), \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot b\\ \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right) \cdot 20}{\left(t\_1 \cdot b\right) \cdot t\_1}, -0.25 \cdot a, \left(-2 \cdot \left(c \cdot c\right)\right) \cdot \frac{c}{t\_1 \cdot \left(b \cdot b\right)}\right), a, \left(-c\right) \cdot \frac{c}{t\_1}\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))) (t_1 (* (* b b) b)))
   (if (<= b 0.027)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (fma
      (fma
       (fma
        (/ (* (* (* (* c c) c) c) 20.0) (* (* t_1 b) t_1))
        (* -0.25 a)
        (* (* -2.0 (* c c)) (/ c (* t_1 (* b b)))))
       a
       (* (- c) (/ c t_1)))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double t_1 = (b * b) * b;
	double tmp;
	if (b <= 0.027) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma(fma(fma((((((c * c) * c) * c) * 20.0) / ((t_1 * b) * t_1)), (-0.25 * a), ((-2.0 * (c * c)) * (c / (t_1 * (b * b))))), a, (-c * (c / t_1))), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_1 = Float64(Float64(b * b) * b)
	tmp = 0.0
	if (b <= 0.027)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = fma(fma(fma(Float64(Float64(Float64(Float64(Float64(c * c) * c) * c) * 20.0) / Float64(Float64(t_1 * b) * t_1)), Float64(-0.25 * a), Float64(Float64(-2.0 * Float64(c * c)) * Float64(c / Float64(t_1 * Float64(b * b))))), a, Float64(Float64(-c) * Float64(c / t_1))), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, 0.027], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * 20.0), $MachinePrecision] / N[(N[(t$95$1 * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * a), $MachinePrecision] + N[(N[(-2.0 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(c / N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[((-c) * N[(c / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0269999999999999997

   1. Initial program 85.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      4. associate-/r* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\color{blue}{\frac{1}{2}}}{a} \]

      6. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{\frac{1}{2}}{a} \]

      7. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot \frac{\frac{1}{2}}{a} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}} \cdot \frac{\frac{1}{2}}{a} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{2}}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot a}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot a} \]

   4. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b} \cdot a}} \]

   6. Applied rewrites 87.1%

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

   if 0.0269999999999999997 < b

   1. Initial program 52.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{\left(\left(-2 \cdot c\right) \cdot c\right) \cdot c}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right), a, \frac{-c}{b}\right)} \]

   7. Applied rewrites 92.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{20 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)}, a \cdot -0.25, \frac{c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)} \cdot \left(\left(c \cdot c\right) \cdot -2\right)\right), a, \left(-c\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right), a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot c\right) \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.25 \cdot a, \left(-2 \cdot \left(c \cdot c\right)\right) \cdot \frac{c}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)}\right), a, \left(-c\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right), a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))