Quadratic roots, narrow range

Percentage Accurate: 55.5% → 99.4%

Time: 11.8s

Alternatives: 10

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: 55.5% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\\ t_1 := \mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - t\_0\right) \cdot b\right)\\ \frac{t\_1 \cdot \frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{2 \cdot a}}{t\_1}}{t\_0 + b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 c) a (* b b))))
        (t_1 (fma (* -4.0 a) c (* (- (* 2.0 b) t_0) b))))
   (/ (* t_1 (/ (/ (fma (* -4.0 c) a 0.0) (* 2.0 a)) t_1)) (+ t_0 b))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * c), a, (b * b)));
	double t_1 = fma((-4.0 * a), c, (((2.0 * b) - t_0) * b));
	return (t_1 * ((fma((-4.0 * c), a, 0.0) / (2.0 * a)) / t_1)) / (t_0 + b);
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))
	t_1 = fma(Float64(-4.0 * a), c, Float64(Float64(Float64(2.0 * b) - t_0) * b))
	return Float64(Float64(t_1 * Float64(Float64(fma(Float64(-4.0 * c), a, 0.0) / Float64(2.0 * a)) / t_1)) / Float64(t_0 + b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(N[(N[(2.0 * b), $MachinePrecision] - t$95$0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(N[(N[(N[(-4.0 * c), $MachinePrecision] * a + 0.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 57.2%

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

4. Applied rewrites 57.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

6. Applied rewrites 58.7%

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

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right) \cdot \frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{2 \cdot a}}{\mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}} \]
9. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\\ \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - t\_0\right)\right)\right) \cdot {\left(\left(t\_0 + b\right) \cdot \left(\mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - t\_0\right) \cdot b\right) \cdot \frac{2 \cdot a}{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}\right)\right)}^{-1} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 c) a (* b b)))))
   (*
    (fma (* -4.0 c) a (fma b b (* b (- b t_0))))
    (pow
     (*
      (+ t_0 b)
      (*
       (fma (* -4.0 a) c (* (- (* 2.0 b) t_0) b))
       (/ (* 2.0 a) (fma (* -4.0 c) a 0.0))))
     -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * c), a, (b * b)));
	return fma((-4.0 * c), a, fma(b, b, (b * (b - t_0)))) * pow(((t_0 + b) * (fma((-4.0 * a), c, (((2.0 * b) - t_0) * b)) * ((2.0 * a) / fma((-4.0 * c), a, 0.0)))), -1.0);
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))
	return Float64(fma(Float64(-4.0 * c), a, fma(b, b, Float64(b * Float64(b - t_0)))) * (Float64(Float64(t_0 + b) * Float64(fma(Float64(-4.0 * a), c, Float64(Float64(Float64(2.0 * b) - t_0) * b)) * Float64(Float64(2.0 * a) / fma(Float64(-4.0 * c), a, 0.0)))) ^ -1.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(b * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$0 + b), $MachinePrecision] * N[(N[(N[(-4.0 * a), $MachinePrecision] * c + N[(N[(N[(2.0 * b), $MachinePrecision] - t$95$0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * a), $MachinePrecision] / N[(N[(-4.0 * c), $MachinePrecision] * a + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.2%

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

4. Applied rewrites 57.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

6. Applied rewrites 58.7%

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

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \color{blue}{\frac{1}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(\mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right) \cdot \frac{2 \cdot a}{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}\right)}} \]

9. Final simplification 99.1%

   \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot {\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(\mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right) \cdot \frac{2 \cdot a}{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}\right)\right)}^{-1} \]
10. Add Preprocessing

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\\ \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - t\_0\right)\right)\right) \cdot \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{\left(t\_0 + b\right) \cdot \left(\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - t\_0\right) \cdot b\right)\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))
	return Float64(fma(Float64(-4.0 * c), a, fma(b, b, Float64(b * Float64(b - t_0)))) * Float64(fma(Float64(-4.0 * c), a, 0.0) / Float64(Float64(t_0 + b) * Float64(Float64(2.0 * a) * fma(Float64(-4.0 * a), c, Float64(Float64(Float64(2.0 * b) - t_0) * b))))))
end

Wolfram

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

Derivation

1. Initial program 57.2%

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

4. Applied rewrites 57.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

6. Applied rewrites 58.7%

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

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)}} \]

9. Final simplification 99.1%

   \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot a, c, \left(2 \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)} \]
10. Add Preprocessing

Alternative 4: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 21:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, c \cdot {b}^{-4}, \frac{a}{\left(-b\right) \cdot b}\right) \cdot c, c, -c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 21.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (/
    (fma
     (* (fma (* (* -2.0 a) a) (* c (pow b -4.0)) (/ a (* (- b) b))) c)
     c
     (- c))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 21.0) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = fma((fma(((-2.0 * a) * a), (c * pow(b, -4.0)), (a / (-b * b))) * c), c, -c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 21.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(-2.0 * a) * a), Float64(c * (b ^ -4.0)), Float64(a / Float64(Float64(-b) * b))) * c), c, Float64(-c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 21.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * a), $MachinePrecision] * N[(c * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] + N[(a / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + (-c)), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 21

   1. Initial program 83.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 84.0

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

   4. Applied rewrites 84.0%

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

   if 21 < b

   1. Initial program 47.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 94.4%

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

   10. Applied rewrites 94.5%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, c \cdot {b}^{-4}, \frac{a}{\left(-b\right) \cdot b}\right) \cdot c, c, -c\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 21:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{b \cdot b}}{b \cdot b} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 21.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (/
    (*
     (fma
      (- (/ (/ (* (* (* a a) c) -2.0) (* b b)) (* b b)) (/ a (* b b)))
      c
      -1.0)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 21.0) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = (fma(((((((a * a) * c) * -2.0) / (b * b)) / (b * b)) - (a / (b * b))), c, -1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 21.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * -2.0) / Float64(b * b)) / Float64(b * b)) - Float64(a / Float64(b * b))), c, -1.0) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 21.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 21

   1. Initial program 83.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 84.0

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

   4. Applied rewrites 84.0%

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

   if 21 < b

   1. Initial program 47.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 94.4%

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

   10. Applied rewrites 94.4%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{b \cdot b}}{b \cdot b} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 170:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-a\right) \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 170.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (/ (- (/ (* (- a) (* c c)) (* b b)) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 170.0) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = (((-a * (c * c)) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 170.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-a) * Float64(c * c)) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 170.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-a) * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 170

   1. Initial program 81.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 81.9

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

   4. Applied rewrites 81.9%

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

   if 170 < b

   1. Initial program 43.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} - c}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 170:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-a\right) \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 170:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-a\right) \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 170.0)
   (* (/ 0.5 a) (- (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/ (- (/ (* (- a) (* c c)) (* b b)) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 170.0) {
		tmp = (0.5 / a) * (sqrt(fma((-4.0 * c), a, (b * b))) - b);
	} else {
		tmp = (((-a * (c * c)) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 170.0)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-a) * Float64(c * c)) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 170.0], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-a) * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 170

   1. Initial program 81.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 81.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 81.7

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

   4. Applied rewrites 81.7%

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

   if 170 < b

   1. Initial program 43.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} - c}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 170:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-a\right) \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

5. Applied rewrites 86.7%

   \[\leadsto \color{blue}{\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 86.6%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b} \]

9. Taylor expanded in a around 0

   \[\leadsto \frac{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} - c}{b} \]
10. Step-by-step derivation

11. Applied rewrites 80.3%

    \[\leadsto \frac{\frac{-a \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b} \]

12. Final simplification 80.3%

    \[\leadsto \frac{\frac{\left(-a\right) \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b} \]
13. Add Preprocessing

Alternative 9: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* (fma (- a) (/ c (* b b)) -1.0) c) b))

C

double code(double a, double b, double c) {
	return (fma(-a, (c / (b * b)), -1.0) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(-a), Float64(c / Float64(b * b)), -1.0) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[((-a) * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

5. Applied rewrites 86.7%

   \[\leadsto \color{blue}{\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 80.2%

   \[\leadsto \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right) \cdot c}{b} \]
9. Add Preprocessing

Alternative 10: 64.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[((-c) / b), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 62.7

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

5. Applied rewrites 62.7%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(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)))