Quadratic roots, narrow range

Percentage Accurate: 55.4% → 98.8%

Time: 15.7s

Alternatives: 15

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: 55.4% 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: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}\\ \frac{a \cdot \mathsf{fma}\left(a \cdot -16, \left(b \cdot b\right) \cdot \left(c \cdot c\right), -4 \cdot \left(c \cdot {b}^{4}\right)\right)}{\left(\left(b \cdot \left(a \cdot 2\right)\right) \cdot t\_0\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, -16 \cdot \left(a \cdot \left(c \cdot c\right)\right), \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot t\_0\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* 4.0 (* a c))))))
   (/
    (* a (fma (* a -16.0) (* (* b b) (* c c)) (* -4.0 (* c (pow b 4.0)))))
    (*
     (* (* b (* a 2.0)) t_0)
     (*
      b
      (+
       (sqrt (fma a (* -16.0 (* a (* c c))) (* (* b b) (* b b))))
       (* b t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (4.0 * (a * c))));
	return (a * fma((a * -16.0), ((b * b) * (c * c)), (-4.0 * (c * pow(b, 4.0))))) / (((b * (a * 2.0)) * t_0) * (b * (sqrt(fma(a, (-16.0 * (a * (c * c))), ((b * b) * (b * b)))) + (b * t_0))));
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(4.0 * Float64(a * c))))
	return Float64(Float64(a * fma(Float64(a * -16.0), Float64(Float64(b * b) * Float64(c * c)), Float64(-4.0 * Float64(c * (b ^ 4.0))))) / Float64(Float64(Float64(b * Float64(a * 2.0)) * t_0) * Float64(b * Float64(sqrt(fma(a, Float64(-16.0 * Float64(a * Float64(c * c))), Float64(Float64(b * b) * Float64(b * b)))) + Float64(b * t_0)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(a * N[(N[(a * -16.0), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(c * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(b * N[(N[Sqrt[N[(a * N[(-16.0 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.4%

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

4. Applied rewrites 52.1%

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-16, a \cdot \left(c \cdot \left(a \cdot c\right)\right), b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} \cdot b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)} \cdot \left(b \cdot b\right)}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)} \cdot b}}}{2 \cdot a} \]

6. Applied rewrites 53.8%

   \[\leadsto \color{blue}{\frac{\left(b \cdot b\right) \cdot \mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) - \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{a \cdot \left(-16 \cdot \left(a \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - 4 \cdot \left({b}^{4} \cdot c\right)\right)}}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{a \cdot \left(-16 \cdot \left(a \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - 4 \cdot \left({b}^{4} \cdot c\right)\right)}}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \frac{a \cdot \color{blue}{\left(-16 \cdot \left(a \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) + \left(\mathsf{neg}\left(4\right)\right) \cdot \left({b}^{4} \cdot c\right)\right)}}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{a \cdot \left(\color{blue}{\left(-16 \cdot a\right) \cdot \left({b}^{2} \cdot {c}^{2}\right)} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left({b}^{4} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{a \cdot \left(\left(-16 \cdot a\right) \cdot \left({b}^{2} \cdot {c}^{2}\right) + \color{blue}{-4} \cdot \left({b}^{4} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{a \cdot \color{blue}{\mathsf{fma}\left(-16 \cdot a, {b}^{2} \cdot {c}^{2}, -4 \cdot \left({b}^{4} \cdot c\right)\right)}}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{a \cdot \mathsf{fma}\left(\color{blue}{-16 \cdot a}, {b}^{2} \cdot {c}^{2}, -4 \cdot \left({b}^{4} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{a \cdot \mathsf{fma}\left(-16 \cdot a, \color{blue}{{b}^{2} \cdot {c}^{2}}, -4 \cdot \left({b}^{4} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{a \cdot \mathsf{fma}\left(-16 \cdot a, \color{blue}{\left(b \cdot b\right)} \cdot {c}^{2}, -4 \cdot \left({b}^{4} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{a \cdot \mathsf{fma}\left(-16 \cdot a, \color{blue}{\left(b \cdot b\right)} \cdot {c}^{2}, -4 \cdot \left({b}^{4} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{a \cdot \mathsf{fma}\left(-16 \cdot a, \left(b \cdot b\right) \cdot \color{blue}{\left(c \cdot c\right)}, -4 \cdot \left({b}^{4} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{a \cdot \mathsf{fma}\left(-16 \cdot a, \left(b \cdot b\right) \cdot \color{blue}{\left(c \cdot c\right)}, -4 \cdot \left({b}^{4} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{a \cdot \mathsf{fma}\left(-16 \cdot a, \left(b \cdot b\right) \cdot \left(c \cdot c\right), \color{blue}{-4 \cdot \left({b}^{4} \cdot c\right)}\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{a \cdot \mathsf{fma}\left(-16 \cdot a, \left(b \cdot b\right) \cdot \left(c \cdot c\right), -4 \cdot \color{blue}{\left({b}^{4} \cdot c\right)}\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

   14. lower-pow.f64 98.9

       \[\leadsto \frac{a \cdot \mathsf{fma}\left(-16 \cdot a, \left(b \cdot b\right) \cdot \left(c \cdot c\right), -4 \cdot \left(\color{blue}{{b}^{4}} \cdot c\right)\right)}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

9. Applied rewrites 98.9%

   \[\leadsto \frac{\color{blue}{a \cdot \mathsf{fma}\left(-16 \cdot a, \left(b \cdot b\right) \cdot \left(c \cdot c\right), -4 \cdot \left({b}^{4} \cdot c\right)\right)}}{\left(\left(\left(a \cdot 2\right) \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(a \cdot \left(c \cdot c\right)\right) \cdot -16, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}\right)\right)} \]

10. Final simplification 98.9%

    \[\leadsto \frac{a \cdot \mathsf{fma}\left(a \cdot -16, \left(b \cdot b\right) \cdot \left(c \cdot c\right), -4 \cdot \left(c \cdot {b}^{4}\right)\right)}{\left(\left(b \cdot \left(a \cdot 2\right)\right) \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(b \cdot \left(\sqrt{\mathsf{fma}\left(a, -16 \cdot \left(a \cdot \left(c \cdot c\right)\right), \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)} + b \cdot \sqrt{\mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}\right)\right)} \]
11. Add Preprocessing

Alternative 2: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot c\right)\\ t_1 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ t_2 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 5:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_1 - b \cdot b\right)}{b + \sqrt{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot t\_0}{b \cdot t\_2} \cdot -2, \frac{-0.25 \cdot \left(\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot 20\right)\right)\right)}{a \cdot \left(t\_2 \cdot t\_2\right)}\right) - \frac{t\_0}{b \cdot b}}{b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c c)))
        (t_1 (fma c (* a -4.0) (* b b)))
        (t_2 (* b (* b b))))
   (if (<= b 5.0)
     (/ (* (/ 0.5 a) (- t_1 (* b b))) (+ b (sqrt t_1)))
     (-
      (/
       (-
        (fma
         a
         (* (/ (* c t_0) (* b t_2)) -2.0)
         (/
          (* -0.25 (* (* a (* a (* a a))) (* c (* (* c (* c c)) 20.0))))
          (* a (* t_2 t_2))))
        (/ t_0 (* b b)))
       b)
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * c);
	double t_1 = fma(c, (a * -4.0), (b * b));
	double t_2 = b * (b * b);
	double tmp;
	if (b <= 5.0) {
		tmp = ((0.5 / a) * (t_1 - (b * b))) / (b + sqrt(t_1));
	} else {
		tmp = ((fma(a, (((c * t_0) / (b * t_2)) * -2.0), ((-0.25 * ((a * (a * (a * a))) * (c * ((c * (c * c)) * 20.0)))) / (a * (t_2 * t_2)))) - (t_0 / (b * b))) / b) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * c))
	t_1 = fma(c, Float64(a * -4.0), Float64(b * b))
	t_2 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 5.0)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_1 - Float64(b * b))) / Float64(b + sqrt(t_1)));
	else
		tmp = Float64(Float64(Float64(fma(a, Float64(Float64(Float64(c * t_0) / Float64(b * t_2)) * -2.0), Float64(Float64(-0.25 * Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(c * Float64(Float64(c * Float64(c * c)) * 20.0)))) / Float64(a * Float64(t_2 * t_2)))) - Float64(t_0 / Float64(b * b))) / b) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 5.0], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * N[(N[(N[(c * t$95$0), $MachinePrecision] / N[(b * t$95$2), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(-0.25 * N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 5

   1. Initial program 80.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 80.5

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

   4. Applied rewrites 80.5%

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

   6. Applied rewrites 82.0%

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

   if 5 < b

   1. Initial program 49.2%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 93.9%

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

   7. Applied rewrites 93.9%

      \[\leadsto \frac{\mathsf{fma}\left(-0.25, \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 20\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]

   9. Applied rewrites 93.9%

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

4. Final simplification 91.5%

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

Reproduce

herbie shell --seed 2024222 
(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)))