Quadratic roots, narrow range

Percentage Accurate: 55.3% → 91.9%

Time: 13.6s

Alternatives: 16

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: 55.3% 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: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - t\_1\right) \cdot b\right)\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{t\_1 + b}}{\left(2 \cdot a\right) \cdot t\_2} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot a, {c}^{4}, {c}^{3} \cdot \left(\left(b \cdot b\right) \cdot -2\right)\right)}{{b}^{7}}, a, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (fma (* -4.0 c) a (fma b b (* (- b t_1) b)))))
   (if (<= b 1.4)
     (* (/ (/ (- t_0 (* b b)) (+ t_1 b)) (* (* 2.0 a) t_2)) t_2)
     (fma
      (fma
       (/
        (fma (* -5.0 a) (pow c 4.0) (* (pow c 3.0) (* (* b b) -2.0)))
        (pow b 7.0))
       a
       (* (/ c (pow b 3.0)) (- c)))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = fma((-4.0 * c), a, fma(b, b, ((b - t_1) * b)));
	double tmp;
	if (b <= 1.4) {
		tmp = (((t_0 - (b * b)) / (t_1 + b)) / ((2.0 * a) * t_2)) * t_2;
	} else {
		tmp = fma(fma((fma((-5.0 * a), pow(c, 4.0), (pow(c, 3.0) * ((b * b) * -2.0))) / pow(b, 7.0)), a, ((c / pow(b, 3.0)) * -c)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - t_1) * b)))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(t_1 + b)) / Float64(Float64(2.0 * a) * t_2)) * t_2);
	else
		tmp = fma(fma(Float64(fma(Float64(-5.0 * a), (c ^ 4.0), Float64((c ^ 3.0) * Float64(Float64(b * b) * -2.0))) / (b ^ 7.0)), a, Float64(Float64(c / (b ^ 3.0)) * Float64(-c))), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(-5.0 * a), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 85.0

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

   6. Applied rewrites 85.0%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-5 \cdot \left(a \cdot {c}^{4}\right) + -2 \cdot \left({b}^{2} \cdot {c}^{3}\right)}{{b}^{7}}, a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 94.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot a, {c}^{4}, \left(-2 \cdot \left(b \cdot b\right)\right) \cdot {c}^{3}\right)}{{b}^{7}}, a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot a, {c}^{4}, {c}^{3} \cdot \left(\left(b \cdot b\right) \cdot -2\right)\right)}{{b}^{7}}, a, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c - \left(a \cdot c\right) \cdot -2\right) \cdot c\\ t_1 := {\left(a \cdot c\right)}^{2}\\ \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{-6}, \frac{{\left(a \cdot c\right)}^{4} \cdot -10}{a} - \mathsf{fma}\left(t\_0 \cdot -2, \mathsf{fma}\left(2, t\_1, t\_1 \cdot -4\right), \left(-8 \cdot {c}^{4}\right) \cdot {a}^{3}\right), \frac{-2}{b \cdot b} \cdot t\_0\right), 0.5, \mathsf{fma}\left(\left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot -0.5\right) \cdot t\_0, {b}^{-4}, -c\right)\right) \cdot {b}^{-3}\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, -2, \mathsf{fma}\left(\frac{2}{b}, \frac{a \cdot a}{b}, \frac{\left({a}^{3} \cdot c\right) \cdot 4}{{b}^{4}}\right) \cdot c\right) \cdot c\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (- (* a c) (* (* a c) -2.0)) c)) (t_1 (pow (* a c) 2.0)))
   (*
    (*
     (fma
      (fma
       (pow b -6.0)
       (-
        (/ (* (pow (* a c) 4.0) -10.0) a)
        (fma
         (* t_0 -2.0)
         (fma 2.0 t_1 (* t_1 -4.0))
         (* (* -8.0 (pow c 4.0)) (pow a 3.0))))
       (* (/ -2.0 (* b b)) t_0))
      0.5
      (fma (* (* (* (* 4.0 a) c) -0.5) t_0) (pow b -4.0) (- c)))
     (pow b -3.0))
    (fma
     b
     b
     (*
      (fma
       a
       -2.0
       (*
        (fma (/ 2.0 b) (/ (* a a) b) (/ (* (* (pow a 3.0) c) 4.0) (pow b 4.0)))
        c))
      c)))))

C

double code(double a, double b, double c) {
	double t_0 = ((a * c) - ((a * c) * -2.0)) * c;
	double t_1 = pow((a * c), 2.0);
	return (fma(fma(pow(b, -6.0), (((pow((a * c), 4.0) * -10.0) / a) - fma((t_0 * -2.0), fma(2.0, t_1, (t_1 * -4.0)), ((-8.0 * pow(c, 4.0)) * pow(a, 3.0)))), ((-2.0 / (b * b)) * t_0)), 0.5, fma(((((4.0 * a) * c) * -0.5) * t_0), pow(b, -4.0), -c)) * pow(b, -3.0)) * fma(b, b, (fma(a, -2.0, (fma((2.0 / b), ((a * a) / b), (((pow(a, 3.0) * c) * 4.0) / pow(b, 4.0))) * c)) * c));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(a * c) - Float64(Float64(a * c) * -2.0)) * c)
	t_1 = Float64(a * c) ^ 2.0
	return Float64(Float64(fma(fma((b ^ -6.0), Float64(Float64(Float64((Float64(a * c) ^ 4.0) * -10.0) / a) - fma(Float64(t_0 * -2.0), fma(2.0, t_1, Float64(t_1 * -4.0)), Float64(Float64(-8.0 * (c ^ 4.0)) * (a ^ 3.0)))), Float64(Float64(-2.0 / Float64(b * b)) * t_0)), 0.5, fma(Float64(Float64(Float64(Float64(4.0 * a) * c) * -0.5) * t_0), (b ^ -4.0), Float64(-c))) * (b ^ -3.0)) * fma(b, b, Float64(fma(a, -2.0, Float64(fma(Float64(2.0 / b), Float64(Float64(a * a) / b), Float64(Float64(Float64((a ^ 3.0) * c) * 4.0) / (b ^ 4.0))) * c)) * c)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(N[Power[b, -6.0], $MachinePrecision] * N[(N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * -10.0), $MachinePrecision] / a), $MachinePrecision] - N[(N[(t$95$0 * -2.0), $MachinePrecision] * N[(2.0 * t$95$1 + N[(t$95$1 * -4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-8.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(N[(N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + (-c)), $MachinePrecision]), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision] * N[(b * b + N[(N[(a * -2.0 + N[(N[(N[(2.0 / b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] * 4.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 55.3%

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

4. Applied rewrites 55.4%

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

5. Taylor expanded in b around inf

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

7. Applied rewrites 90.6%

   \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(-0.5, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), -0.5 \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}}} \]

8. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\left(c \cdot \left(-4 \cdot a + \left(2 \cdot a + c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)\right) + {b}^{2}\right)} \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({b}^{2} + c \cdot \left(-4 \cdot a + \left(2 \cdot a + c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)\right)\right)} \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

   2. unpow2 N/A

      \[\leadsto \left(\color{blue}{b \cdot b} + c \cdot \left(-4 \cdot a + \left(2 \cdot a + c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)\right)\right) \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a + \left(2 \cdot a + c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)\right)\right)} \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a + \left(2 \cdot a + c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)\right)}\right) \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

   5. associate-+r+ N/A

      \[\leadsto \mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4 \cdot a + 2 \cdot a\right) + c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)}\right) \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{a \cdot \left(-4 + 2\right)} + c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)\right) \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-2} + c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)\right) \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(b, b, c \cdot \color{blue}{\mathsf{fma}\left(a, -2, c \cdot \left(2 \cdot \frac{{a}^{2}}{{b}^{2}} + 4 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}}\right)\right)}\right) \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), \frac{-1}{2} \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

10. Applied rewrites 90.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, c \cdot \mathsf{fma}\left(a, -2, c \cdot \mathsf{fma}\left(\frac{2}{b}, \frac{a \cdot a}{b}, \frac{4 \cdot \left({a}^{3} \cdot c\right)}{{b}^{4}}\right)\right)\right)} \cdot \frac{\mathsf{fma}\left(-1, c, \mathsf{fma}\left(0.5, \mathsf{fma}\left(\frac{-2}{b}, \frac{a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)}{b}, \frac{\mathsf{fma}\left(-0.5, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{a}, -\mathsf{fma}\left(-8 \cdot {a}^{3}, {c}^{4}, \mathsf{fma}\left(2 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right), -{\left(\left(a \cdot c\right) \cdot -2\right)}^{2} \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right), -0.5 \cdot \frac{\left(\left(a \cdot c\right) \cdot -2\right) \cdot \left(-2 \cdot \left(a \cdot \left(c \cdot c\right) - c \cdot \left(\left(a \cdot c\right) \cdot -2\right)\right)\right)}{{b}^{4}}\right)\right)}{{b}^{3}} \]

12. Applied rewrites 90.9%

    \[\leadsto \mathsf{fma}\left(b, b, c \cdot \mathsf{fma}\left(a, -2, c \cdot \mathsf{fma}\left(\frac{2}{b}, \frac{a \cdot a}{b}, \frac{4 \cdot \left({a}^{3} \cdot c\right)}{{b}^{4}}\right)\right)\right) \cdot \color{blue}{\left({b}^{-3} \cdot \mathsf{fma}\left(\mathsf{fma}\left({b}^{-6}, \frac{-10 \cdot {\left(c \cdot a\right)}^{4}}{a} - \mathsf{fma}\left(\left(c \cdot \left(c \cdot a - -2 \cdot \left(c \cdot a\right)\right)\right) \cdot -2, \mathsf{fma}\left(2, {\left(c \cdot a\right)}^{2}, {\left(c \cdot a\right)}^{2} \cdot -4\right), \left({c}^{4} \cdot -8\right) \cdot {a}^{3}\right), \left(c \cdot \left(c \cdot a - -2 \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{-2}{b \cdot b}\right), 0.5, \mathsf{fma}\left(\left(-0.5 \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(c \cdot \left(c \cdot a - -2 \cdot \left(c \cdot a\right)\right)\right), {b}^{-4}, -c\right)\right)\right)} \]

13. Final simplification 90.9%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left({b}^{-6}, \frac{{\left(a \cdot c\right)}^{4} \cdot -10}{a} - \mathsf{fma}\left(\left(\left(a \cdot c - \left(a \cdot c\right) \cdot -2\right) \cdot c\right) \cdot -2, \mathsf{fma}\left(2, {\left(a \cdot c\right)}^{2}, {\left(a \cdot c\right)}^{2} \cdot -4\right), \left(-8 \cdot {c}^{4}\right) \cdot {a}^{3}\right), \frac{-2}{b \cdot b} \cdot \left(\left(a \cdot c - \left(a \cdot c\right) \cdot -2\right) \cdot c\right)\right), 0.5, \mathsf{fma}\left(\left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot -0.5\right) \cdot \left(\left(a \cdot c - \left(a \cdot c\right) \cdot -2\right) \cdot c\right), {b}^{-4}, -c\right)\right) \cdot {b}^{-3}\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, -2, \mathsf{fma}\left(\frac{2}{b}, \frac{a \cdot a}{b}, \frac{\left({a}^{3} \cdot c\right) \cdot 4}{{b}^{4}}\right) \cdot c\right) \cdot c\right) \]
14. Add Preprocessing

Alternative 3: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{20 \cdot \frac{{a}^{4}}{{b}^{6}}}{b} \cdot c, -0.5, \frac{-4 \cdot {a}^{3}}{{b}^{5}}\right), c, \frac{\left(a \cdot a\right) \cdot -2}{{b}^{3}}\right), c, \frac{a}{b} \cdot -2\right) \cdot c}{\left(2 \cdot a\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, -2, \mathsf{fma}\left(\frac{2}{b}, \frac{a \cdot a}{b}, \frac{\left({a}^{3} \cdot c\right) \cdot 4}{{b}^{4}}\right) \cdot c\right) \cdot c\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/
   (*
    (fma
     (fma
      (fma
       (* (/ (* 20.0 (/ (pow a 4.0) (pow b 6.0))) b) c)
       -0.5
       (/ (* -4.0 (pow a 3.0)) (pow b 5.0)))
      c
      (/ (* (* a a) -2.0) (pow b 3.0)))
     c
     (* (/ a b) -2.0))
    c)
   (*
    (* 2.0 a)
    (fma
     b
     b
     (*
      (fma
       a
       -2.0
       (*
        (fma (/ 2.0 b) (/ (* a a) b) (/ (* (* (pow a 3.0) c) 4.0) (pow b 4.0)))
        c))
      c))))
  (fma (* -4.0 c) a (fma b b (* (- b (sqrt (fma (* -4.0 c) a (* b b)))) b)))))

C

double code(double a, double b, double c) {
	return ((fma(fma(fma((((20.0 * (pow(a, 4.0) / pow(b, 6.0))) / b) * c), -0.5, ((-4.0 * pow(a, 3.0)) / pow(b, 5.0))), c, (((a * a) * -2.0) / pow(b, 3.0))), c, ((a / b) * -2.0)) * c) / ((2.0 * a) * fma(b, b, (fma(a, -2.0, (fma((2.0 / b), ((a * a) / b), (((pow(a, 3.0) * c) * 4.0) / pow(b, 4.0))) * c)) * c)))) * fma((-4.0 * c), a, fma(b, b, ((b - sqrt(fma((-4.0 * c), a, (b * b)))) * b)));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(fma(fma(fma(Float64(Float64(Float64(20.0 * Float64((a ^ 4.0) / (b ^ 6.0))) / b) * c), -0.5, Float64(Float64(-4.0 * (a ^ 3.0)) / (b ^ 5.0))), c, Float64(Float64(Float64(a * a) * -2.0) / (b ^ 3.0))), c, Float64(Float64(a / b) * -2.0)) * c) / Float64(Float64(2.0 * a) * fma(b, b, Float64(fma(a, -2.0, Float64(fma(Float64(2.0 / b), Float64(Float64(a * a) / b), Float64(Float64(Float64((a ^ 3.0) * c) * 4.0) / (b ^ 4.0))) * c)) * c)))) * fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))) * b))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(20.0 * N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision] * -0.5 + N[(N[(-4.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(N[(a / b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * N[(b * b + N[(N[(a * -2.0 + N[(N[(N[(2.0 / b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] * 4.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

4. Applied rewrites 55.4%

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

5. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. associate-+r+ N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 47.8%

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

8. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \frac{\color{blue}{c \cdot \left(-2 \cdot \frac{a}{b} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-4 \cdot \frac{{a}^{3}}{{b}^{5}} + \frac{-1}{2} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{\mathsf{fma}\left(b, b, c \cdot \mathsf{fma}\left(a, -2, c \cdot \mathsf{fma}\left(\frac{2}{b}, \frac{a \cdot a}{b}, \frac{4 \cdot \left({a}^{3} \cdot c\right)}{{b}^{4}}\right)\right)\right) \cdot \left(2 \cdot a\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 90.9%

    \[\leadsto \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{b}, -0.5, \frac{-4 \cdot {a}^{3}}{{b}^{5}}\right), c, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{3}}\right), c, \frac{a}{b} \cdot -2\right) \cdot c}}{\mathsf{fma}\left(b, b, c \cdot \mathsf{fma}\left(a, -2, c \cdot \mathsf{fma}\left(\frac{2}{b}, \frac{a \cdot a}{b}, \frac{4 \cdot \left({a}^{3} \cdot c\right)}{{b}^{4}}\right)\right)\right) \cdot \left(2 \cdot a\right)} \]

11. Final simplification 90.9%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{20 \cdot \frac{{a}^{4}}{{b}^{6}}}{b} \cdot c, -0.5, \frac{-4 \cdot {a}^{3}}{{b}^{5}}\right), c, \frac{\left(a \cdot a\right) \cdot -2}{{b}^{3}}\right), c, \frac{a}{b} \cdot -2\right) \cdot c}{\left(2 \cdot a\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, -2, \mathsf{fma}\left(\frac{2}{b}, \frac{a \cdot a}{b}, \frac{\left({a}^{3} \cdot c\right) \cdot 4}{{b}^{4}}\right) \cdot c\right) \cdot c\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right) \]
12. Add Preprocessing

Alternative 4: 89.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - t\_1\right) \cdot b\right)\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{t\_1 + b}}{\left(2 \cdot a\right) \cdot t\_2} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{{c}^{3}}{{b}^{5}}, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (fma (* -4.0 c) a (fma b b (* (- b t_1) b)))))
   (if (<= b 1.4)
     (* (/ (/ (- t_0 (* b b)) (+ t_1 b)) (* (* 2.0 a) t_2)) t_2)
     (fma
      (fma (* -2.0 a) (/ (pow c 3.0) (pow b 5.0)) (* (/ c (pow b 3.0)) (- c)))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = fma((-4.0 * c), a, fma(b, b, ((b - t_1) * b)));
	double tmp;
	if (b <= 1.4) {
		tmp = (((t_0 - (b * b)) / (t_1 + b)) / ((2.0 * a) * t_2)) * t_2;
	} else {
		tmp = fma(fma((-2.0 * a), (pow(c, 3.0) / pow(b, 5.0)), ((c / pow(b, 3.0)) * -c)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - t_1) * b)))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(t_1 + b)) / Float64(Float64(2.0 * a) * t_2)) * t_2);
	else
		tmp = fma(fma(Float64(-2.0 * a), Float64((c ^ 3.0) / (b ^ 5.0)), Float64(Float64(c / (b ^ 3.0)) * Float64(-c))), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(-2.0 * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 85.0

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

   6. Applied rewrites 85.0%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 91.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot a, \frac{{c}^{3}}{{b}^{5}}, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - t\_1\right) \cdot b\right)\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{t\_1 + b}}{\left(2 \cdot a\right) \cdot t\_2} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \mathsf{fma}\left(2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{4}}, \frac{c \cdot c}{b} \cdot \frac{a}{b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (fma (* -4.0 c) a (fma b b (* (- b t_1) b)))))
   (if (<= b 1.4)
     (* (/ (/ (- t_0 (* b b)) (+ t_1 b)) (* (* 2.0 a) t_2)) t_2)
     (/
      (-
       (- c)
       (fma
        2.0
        (/ (* (pow c 3.0) (* a a)) (pow b 4.0))
        (* (/ (* c c) b) (/ a b))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = fma((-4.0 * c), a, fma(b, b, ((b - t_1) * b)));
	double tmp;
	if (b <= 1.4) {
		tmp = (((t_0 - (b * b)) / (t_1 + b)) / ((2.0 * a) * t_2)) * t_2;
	} else {
		tmp = (-c - fma(2.0, ((pow(c, 3.0) * (a * a)) / pow(b, 4.0)), (((c * c) / b) * (a / b)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - t_1) * b)))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(t_1 + b)) / Float64(Float64(2.0 * a) * t_2)) * t_2);
	else
		tmp = Float64(Float64(Float64(-c) - fma(2.0, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 4.0)), Float64(Float64(Float64(c * c) / b) * Float64(a / b)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[((-c) - N[(2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 85.0

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

   6. Applied rewrites 85.0%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 91.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \mathsf{fma}\left(2, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{4}}, \frac{c \cdot c}{b} \cdot \frac{a}{b}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - t\_1\right) \cdot b\right)\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{t\_1 + b}}{\left(2 \cdot a\right) \cdot t\_2} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (fma (* -4.0 c) a (fma b b (* (- b t_1) b)))))
   (if (<= b 1.4)
     (* (/ (/ (- t_0 (* b b)) (+ t_1 b)) (* (* 2.0 a) t_2)) t_2)
     (/
      (-
       (/ (* (* (* (pow c 3.0) a) a) -2.0) (pow b 4.0))
       (fma (/ c b) (/ (* a c) b) c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = fma((-4.0 * c), a, fma(b, b, ((b - t_1) * b)));
	double tmp;
	if (b <= 1.4) {
		tmp = (((t_0 - (b * b)) / (t_1 + b)) / ((2.0 * a) * t_2)) * t_2;
	} else {
		tmp = (((((pow(c, 3.0) * a) * a) * -2.0) / pow(b, 4.0)) - fma((c / b), ((a * c) / b), c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - t_1) * b)))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(t_1 + b)) / Float64(Float64(2.0 * a) * t_2)) * t_2);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64((c ^ 3.0) * a) * a) * -2.0) / (b ^ 4.0)) - fma(Float64(c / b), Float64(Float64(a * c) / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 85.0

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

   6. Applied rewrites 85.0%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 91.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - t\_1\right) \cdot b\right)\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{t\_1 + b}}{\left(2 \cdot a\right) \cdot t\_2} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c}{{b}^{5}}, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (fma (* -4.0 c) a (fma b b (* (- b t_1) b)))))
   (if (<= b 1.4)
     (* (/ (/ (- t_0 (* b b)) (+ t_1 b)) (* (* 2.0 a) t_2)) t_2)
     (*
      (fma
       (fma (* (* -2.0 a) a) (/ c (pow b 5.0)) (/ (- a) (pow b 3.0)))
       c
       (/ -1.0 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = fma((-4.0 * c), a, fma(b, b, ((b - t_1) * b)));
	double tmp;
	if (b <= 1.4) {
		tmp = (((t_0 - (b * b)) / (t_1 + b)) / ((2.0 * a) * t_2)) * t_2;
	} else {
		tmp = fma(fma(((-2.0 * a) * a), (c / pow(b, 5.0)), (-a / pow(b, 3.0))), c, (-1.0 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - t_1) * b)))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(t_1 + b)) / Float64(Float64(2.0 * a) * t_2)) * t_2);
	else
		tmp = Float64(fma(fma(Float64(Float64(-2.0 * a) * a), Float64(c / (b ^ 5.0)), Float64(Float64(-a) / (b ^ 3.0))), c, Float64(-1.0 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * a), $MachinePrecision] * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[((-a) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 85.0

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

   6. Applied rewrites 85.0%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(a \cdot -2\right) \cdot a, \frac{c}{{b}^{5}}, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{c}{{b}^{5}}, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - t\_1\right) \cdot b\right)\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{t\_1 + b}}{\left(2 \cdot a\right) \cdot t\_2} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (fma (* -4.0 c) a (fma b b (* (- b t_1) b)))))
   (if (<= b 1.4)
     (* (/ (/ (- t_0 (* b b)) (+ t_1 b)) (* (* 2.0 a) t_2)) t_2)
     (/ (fma (/ c b) (/ (* a c) b) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = fma((-4.0 * c), a, fma(b, b, ((b - t_1) * b)));
	double tmp;
	if (b <= 1.4) {
		tmp = (((t_0 - (b * b)) / (t_1 + b)) / ((2.0 * a) * t_2)) * t_2;
	} else {
		tmp = fma((c / b), ((a * c) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - t_1) * b)))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(t_1 + b)) / Float64(Float64(2.0 * a) * t_2)) * t_2);
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(a * c) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 85.0

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

   6. Applied rewrites 85.0%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{\left(2 \cdot a\right) \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\mathsf{fma}\left(\left(b + b\right) - t\_1, b, \left(a \cdot c\right) \cdot -4\right) \cdot \left(2 \cdot a\right)\right) \cdot \left(t\_1 + b\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - t\_1\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 1.4)
     (*
      (/
       (- t_0 (* b b))
       (* (* (fma (- (+ b b) t_1) b (* (* a c) -4.0)) (* 2.0 a)) (+ t_1 b)))
      (fma (* -4.0 c) a (fma b b (* (- b t_1) b))))
     (/ (fma (/ c b) (/ (* a c) b) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 1.4) {
		tmp = ((t_0 - (b * b)) / ((fma(((b + b) - t_1), b, ((a * c) * -4.0)) * (2.0 * a)) * (t_1 + b))) * fma((-4.0 * c), a, fma(b, b, ((b - t_1) * b)));
	} else {
		tmp = fma((c / b), ((a * c) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(fma(Float64(Float64(b + b) - t_1), b, Float64(Float64(a * c) * -4.0)) * Float64(2.0 * a)) * Float64(t_1 + b))) * fma(Float64(-4.0 * c), a, fma(b, b, Float64(Float64(b - t_1) * b))));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(a * c) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(b + b), $MachinePrecision] - t$95$1), $MachinePrecision] * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(N[(b - t$95$1), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, b \cdot \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)\right)\right) \cdot \left(2 \cdot a\right)}} \]
   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. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 84.8%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\mathsf{fma}\left(\left(b + b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}, b, \left(a \cdot c\right) \cdot -4\right) \cdot \left(2 \cdot a\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \left(b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -2.52 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* 2.0 a)) -2.52e-6)
   (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (2.0 * a)) <= -2.52e-6) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(2.0 * a)) <= -2.52e-6)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -2.52e-6], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2.52000000000000001e-6

   1. Initial program 72.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 72.6

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

      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 72.6

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

   4. Applied rewrites 72.6%

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

   if -2.52000000000000001e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 31.7%

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

   3. Taylor expanded in c 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 84.1

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

   5. Applied rewrites 84.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a} \leq -2.52 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{0.5}{-a} \cdot \mathsf{fma}\left(b, b, -t\_0\right)}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= b 1.4)
     (/ (* (/ 0.5 (- a)) (fma b b (- t_0))) (+ (sqrt t_0) b))
     (/ (fma (/ c b) (/ (* a c) b) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (b <= 1.4) {
		tmp = ((0.5 / -a) * fma(b, b, -t_0)) / (sqrt(t_0) + b);
	} else {
		tmp = fma((c / b), ((a * c) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(Float64(0.5 / Float64(-a)) * fma(b, b, Float64(-t_0))) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(a * c) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(N[(0.5 / (-a)), $MachinePrecision] * N[(b * b + (-t$95$0)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 83.8%

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

   8. Applied rewrites 84.6%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{0.5}{-a} \cdot \mathsf{fma}\left(b, b, -\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -t\_0\right)}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= b 1.4)
     (/ (fma b b (- t_0)) (* (- (- b) (sqrt t_0)) (* 2.0 a)))
     (/ (fma (/ c b) (/ (* a c) b) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (b <= 1.4) {
		tmp = fma(b, b, -t_0) / ((-b - sqrt(t_0)) * (2.0 * a));
	} else {
		tmp = fma((c / b), ((a * c) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(fma(b, b, Float64(-t_0)) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(2.0 * a)));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(a * c) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(N[(b * b + (-t$95$0)), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

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

   4. Applied rewrites 83.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 83.8%

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

   8. Applied rewrites 84.5%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.4)
   (/ (- (sqrt (fma b b (* (* -4.0 c) a))) b) (* 2.0 a))
   (/ (fma (/ c b) (/ (* a c) b) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4) {
		tmp = (sqrt(fma(b, b, ((-4.0 * c) * a))) - b) / (2.0 * a);
	} else {
		tmp = fma((c / b), ((a * c) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(a * c) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.4], N[(N[(N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 83.2

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

   4. Applied rewrites 83.2%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.4)
   (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
   (/ (fma (/ c b) (/ (* a c) b) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = fma((c / b), ((a * c) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(a * c) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.4], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 83.0

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

      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 83.0

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

   4. Applied rewrites 83.1%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{a \cdot c}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b} \cdot \left(-c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.4)
   (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
   (* (/ (fma (/ a b) (/ c b) 1.0) b) (- c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = (fma((a / b), (c / b), 1.0) / b) * -c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(fma(Float64(a / b), Float64(c / b), 1.0) / b) * Float64(-c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.4], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision] * (-c)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 83.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 83.0

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

      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 83.0

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

   4. Applied rewrites 83.1%

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

   if 1.3999999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in c around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto -c \cdot \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{3}}, a, \frac{1}{b}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto -c \cdot \mathsf{fma}\left(\color{blue}{\frac{c}{{b}^{3}}}, a, \frac{1}{b}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto -c \cdot \mathsf{fma}\left(\frac{c}{\color{blue}{{b}^{3}}}, a, \frac{1}{b}\right) \]

      12. lower-/.f64 86.6

          \[\leadsto -c \cdot \mathsf{fma}\left(\frac{c}{{b}^{3}}, a, \color{blue}{\frac{1}{b}}\right) \]

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{-c \cdot \mathsf{fma}\left(\frac{c}{{b}^{3}}, a, \frac{1}{b}\right)} \]

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 86.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b} \cdot \left(-c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.5% 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 55.3%

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

3. Taylor expanded in c 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 64.5

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

5. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2024284 
(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)))