Quadratic roots, narrow range

Percentage Accurate: 55.5% → 91.9%

Time: 12.7s

Alternatives: 18

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{4}}\\ t_1 := \left(4 \cdot a\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - t\_1}}{2 \cdot a} \leq -49:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, t\_1\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{b \cdot \left(\left({\left(b \cdot b\right)}^{-1} - \frac{\mathsf{fma}\left(c \cdot c, -3, c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot t\_0, \mathsf{fma}\left(a, t\_0, \frac{{a}^{-1}}{c}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 4.0))) (t_1 (* (* 4.0 a) c)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) t_1))) (* 2.0 a)) -49.0)
     (/
      (* (fma b b (fma (- b) b t_1)) (/ 0.5 (- a)))
      (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
     (/
      (pow a -1.0)
      (*
       b
       (-
        (-
         (pow (* b b) -1.0)
         (/ (* (fma (* c c) -3.0 (* c c)) (* a a)) (pow b 6.0)))
        (fma -2.0 (* a t_0) (fma a t_0 (/ (pow a -1.0) c)))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 4.0);
	double t_1 = (4.0 * a) * c;
	double tmp;
	if (((-b + sqrt(((b * b) - t_1))) / (2.0 * a)) <= -49.0) {
		tmp = (fma(b, b, fma(-b, b, t_1)) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = pow(a, -1.0) / (b * ((pow((b * b), -1.0) - ((fma((c * c), -3.0, (c * c)) * (a * a)) / pow(b, 6.0))) - fma(-2.0, (a * t_0), fma(a, t_0, (pow(a, -1.0) / c)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 4.0))
	t_1 = Float64(Float64(4.0 * a) * c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - t_1))) / Float64(2.0 * a)) <= -49.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, t_1)) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(b * Float64(Float64((Float64(b * b) ^ -1.0) - Float64(Float64(fma(Float64(c * c), -3.0, Float64(c * c)) * Float64(a * a)) / (b ^ 6.0))) - fma(-2.0, Float64(a * t_0), fma(a, t_0, Float64((a ^ -1.0) / c))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -49.0], N[(N[(N[(b * b + N[((-b) * b + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(b * N[(N[(N[Power[N[(b * b), $MachinePrecision], -1.0], $MachinePrecision] - N[(N[(N[(N[(c * c), $MachinePrecision] * -3.0 + N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(a * t$95$0), $MachinePrecision] + N[(a * t$95$0 + N[(N[Power[a, -1.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -49

   1. Initial program 89.6%

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

   4. Applied rewrites 89.5%

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

   6. Applied rewrites 88.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 88.4%

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

   10. Applied rewrites 91.0%

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

   if -49 < (/.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 51.0%

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 94.2%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, c \cdot \left(-1 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 94.2%

       \[\leadsto \frac{{a}^{-1}}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(c \cdot c, -3, c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -49:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{b \cdot \left(\left({\left(b \cdot b\right)}^{-1} - \frac{\mathsf{fma}\left(c \cdot c, -3, c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{{a}^{-1}}{c}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot a\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - t\_0}}{2 \cdot a} \leq -49:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, t\_0\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{5}}, 2, \frac{a}{{b}^{3}}\right), c, {b}^{-1}\right), c, \frac{-b}{a}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* 4.0 a) c)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) t_0))) (* 2.0 a)) -49.0)
     (/
      (* (fma b b (fma (- b) b t_0)) (/ 0.5 (- a)))
      (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
     (/
      (pow a -1.0)
      (/
       (fma
        (fma
         (fma (* (* a a) (/ c (pow b 5.0))) 2.0 (/ a (pow b 3.0)))
         c
         (pow b -1.0))
        c
        (/ (- b) a))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = (4.0 * a) * c;
	double tmp;
	if (((-b + sqrt(((b * b) - t_0))) / (2.0 * a)) <= -49.0) {
		tmp = (fma(b, b, fma(-b, b, t_0)) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = pow(a, -1.0) / (fma(fma(fma(((a * a) * (c / pow(b, 5.0))), 2.0, (a / pow(b, 3.0))), c, pow(b, -1.0)), c, (-b / a)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(4.0 * a) * c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - t_0))) / Float64(2.0 * a)) <= -49.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, t_0)) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(fma(fma(fma(Float64(Float64(a * a) * Float64(c / (b ^ 5.0))), 2.0, Float64(a / (b ^ 3.0))), c, (b ^ -1.0)), c, Float64(Float64(-b) / a)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -49.0], N[(N[(N[(b * b + N[((-b) * b + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * c + N[((-b) / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $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)) < -49

   1. Initial program 89.6%

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

   4. Applied rewrites 89.5%

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

   6. Applied rewrites 88.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 88.4%

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

   10. Applied rewrites 91.0%

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

   if -49 < (/.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 51.0%

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

   4. Applied rewrites 51.0%

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

   6. Applied rewrites 51.9%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 94.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -49:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{5}}, 2, \frac{a}{{b}^{3}}\right), c, {b}^{-1}\right), c, \frac{-b}{a}\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(a, \frac{c}{{b}^{3}}, {b}^{-1}\right), a, \frac{-b}{c}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (* (fma b b (fma (- b) b (* (* 4.0 a) c))) (/ 0.5 (- a)))
    (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/
    (pow a -1.0)
    (/ (fma (fma a (/ c (pow b 3.0)) (pow b -1.0)) a (/ (- b) c)) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (fma(b, b, fma(-b, b, ((4.0 * a) * c))) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = pow(a, -1.0) / (fma(fma(a, (c / pow(b, 3.0)), pow(b, -1.0)), a, (-b / c)) / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c))) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(fma(fma(a, Float64(c / (b ^ 3.0)), (b ^ -1.0)), a, Float64(Float64(-b) / c)) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * a + N[((-b) / c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 85.0%

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

   if 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   6. Applied rewrites 44.0%

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

   7. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. mul-1-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 95.0

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

   9. Applied rewrites 95.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(a, \frac{c}{{b}^{3}}, {b}^{-1}\right), a, \frac{-b}{c}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot a\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - t\_0}}{2 \cdot a} \leq -49:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, t\_0\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-5 \cdot {a}^{3}, c, \left(\left(a \cdot a\right) \cdot -2\right) \cdot \left(b \cdot b\right)\right)}{{b}^{6}}, \frac{-a}{b \cdot b}\right), -1\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* 4.0 a) c)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) t_0))) (* 2.0 a)) -49.0)
     (/
      (* (fma b b (fma (- b) b t_0)) (/ 0.5 (- a)))
      (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
     (/
      (*
       c
       (fma
        c
        (fma
         c
         (/
          (fma (* -5.0 (pow a 3.0)) c (* (* (* a a) -2.0) (* b b)))
          (pow b 6.0))
         (/ (- a) (* b b)))
        -1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = (4.0 * a) * c;
	double tmp;
	if (((-b + sqrt(((b * b) - t_0))) / (2.0 * a)) <= -49.0) {
		tmp = (fma(b, b, fma(-b, b, t_0)) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = (c * fma(c, fma(c, (fma((-5.0 * pow(a, 3.0)), c, (((a * a) * -2.0) * (b * b))) / pow(b, 6.0)), (-a / (b * b))), -1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(4.0 * a) * c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - t_0))) / Float64(2.0 * a)) <= -49.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, t_0)) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64(Float64(c * fma(c, fma(c, Float64(fma(Float64(-5.0 * (a ^ 3.0)), c, Float64(Float64(Float64(a * a) * -2.0) * Float64(b * b))) / (b ^ 6.0)), Float64(Float64(-a) / Float64(b * b))), -1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -49.0], N[(N[(N[(b * b + N[((-b) * b + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(c * N[(N[(N[(-5.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] * c + N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[((-a) / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / 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)) < -49

   1. Initial program 89.6%

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

   4. Applied rewrites 89.5%

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

   6. Applied rewrites 88.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 88.4%

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

   10. Applied rewrites 91.0%

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

   if -49 < (/.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 51.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.1%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 94.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 94.0%

       \[\leadsto \frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-5 \cdot {a}^{3}, c, \left(\left(a \cdot a\right) \cdot -2\right) \cdot \left(b \cdot b\right)\right)}{{b}^{6}}, -\frac{a}{b \cdot b}\right), -1\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -49:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-5 \cdot {a}^{3}, c, \left(\left(a \cdot a\right) \cdot -2\right) \cdot \left(b \cdot b\right)\right)}{{b}^{6}}, \frac{-a}{b \cdot b}\right), -1\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{b \cdot \left({\left(b \cdot b\right)}^{-1} - \frac{{a}^{-1}}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (* (fma b b (fma (- b) b (* (* 4.0 a) c))) (/ 0.5 (- a)))
    (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/ (pow a -1.0) (* b (- (pow (* b b) -1.0) (/ (pow a -1.0) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (fma(b, b, fma(-b, b, ((4.0 * a) * c))) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = pow(a, -1.0) / (b * (pow((b * b), -1.0) - (pow(a, -1.0) / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c))) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(b * Float64((Float64(b * b) ^ -1.0) - Float64((a ^ -1.0) / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(b * N[(N[Power[N[(b * b), $MachinePrecision], -1.0], $MachinePrecision] - N[(N[Power[a, -1.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 85.0%

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

   if 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 90.7

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

   7. Applied rewrites 90.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{b \cdot \left({\left(b \cdot b\right)}^{-1} - \frac{{a}^{-1}}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{4}} - \frac{c}{b} \cdot \frac{c}{b}, -c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (* (fma b b (fma (- b) b (* (* 4.0 a) c))) (/ 0.5 (- a)))
    (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/
    (fma
     a
     (- (/ (* -2.0 (* a (pow c 3.0))) (pow b 4.0)) (* (/ c b) (/ c b)))
     (- c))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (fma(b, b, fma(-b, b, ((4.0 * a) * c))) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = fma(a, (((-2.0 * (a * pow(c, 3.0))) / pow(b, 4.0)) - ((c / b) * (c / b))), -c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c))) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(-2.0 * Float64(a * (c ^ 3.0))) / (b ^ 4.0)) - Float64(Float64(c / b) * Float64(c / b))), Float64(-c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(-2.0 * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 85.0%

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

   if 37 < b

   1. Initial program 43.3%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{4}} - \frac{c}{b} \cdot \frac{c}{b}, -c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot c, a \cdot \frac{a}{{b}^{5}}, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (* (fma b b (fma (- b) b (* (* 4.0 a) c))) (/ 0.5 (- a)))
    (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
   (*
    (fma
     (fma (* -2.0 c) (* a (/ a (pow b 5.0))) (/ (- a) (pow b 3.0)))
     c
     (/ -1.0 b))
    c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (fma(b, b, fma(-b, b, ((4.0 * a) * c))) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = fma(fma((-2.0 * c), (a * (a / pow(b, 5.0))), (-a / pow(b, 3.0))), c, (-1.0 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c))) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64(fma(fma(Float64(-2.0 * c), Float64(a * Float64(a / (b ^ 5.0))), Float64(Float64(-a) / (b ^ 3.0))), c, Float64(-1.0 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * c), $MachinePrecision] * N[(a * N[(a / N[Power[b, 5.0], $MachinePrecision]), $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 < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 85.0%

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

   if 37 < b

   1. Initial program 43.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
   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 95.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot c, a \cdot \frac{a}{{b}^{5}}, \frac{-a}{{b}^{3}}\right), c, \frac{-1}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{b \cdot \frac{\frac{c}{b \cdot b} - {a}^{-1}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (* (fma b b (fma (- b) b (* (* 4.0 a) c))) (/ 0.5 (- a)))
    (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/ (pow a -1.0) (* b (/ (- (/ c (* b b)) (pow a -1.0)) c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (fma(b, b, fma(-b, b, ((4.0 * a) * c))) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = pow(a, -1.0) / (b * (((c / (b * b)) - pow(a, -1.0)) / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c))) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(b * Float64(Float64(Float64(c / Float64(b * b)) - (a ^ -1.0)) / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(b * N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 85.0%

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

   if 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 96.8%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, c \cdot \left(-1 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)}} \]

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 90.6%

       \[\leadsto \frac{{a}^{-1}}{b \cdot \frac{\frac{c}{b \cdot b} - \frac{1}{a}}{\color{blue}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{b \cdot \frac{\frac{c}{b \cdot b} - {a}^{-1}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c, \frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}} - \frac{a}{b \cdot b}, -1\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (* (fma b b (fma (- b) b (* (* 4.0 a) c))) (/ 0.5 (- a)))
    (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/
    (* c (fma c (- (/ (* -2.0 (* (* a a) c)) (pow b 4.0)) (/ a (* b b))) -1.0))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (fma(b, b, fma(-b, b, ((4.0 * a) * c))) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = (c * fma(c, (((-2.0 * ((a * a) * c)) / pow(b, 4.0)) - (a / (b * b))), -1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c))) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64(Float64(c * fma(c, Float64(Float64(Float64(-2.0 * Float64(Float64(a * a) * c)) / (b ^ 4.0)) - Float64(a / Float64(b * b))), -1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(N[(N[(-2.0 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 85.0%

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

   if 37 < b

   1. Initial program 43.3%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 95.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c, \frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}} - \frac{a}{b \cdot b}, -1\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (* (fma b b (fma (- b) b (* (* 4.0 a) c))) (/ 0.5 (- a)))
    (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/ (pow a -1.0) (/ (- (/ a b) (/ b c)) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (fma(b, b, fma(-b, b, ((4.0 * a) * c))) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = pow(a, -1.0) / (((a / b) - (b / c)) / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c))) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(a / b) - Float64(b / c)) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 85.0%

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

   if 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (* (fma b b (fma (- b) b (* (* 4.0 a) c))) (/ 0.5 (- a)))
    (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/ (- (- c) (/ (* a (* c c)) (* b b))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (fma(b, b, fma(-b, b, ((4.0 * a) * c))) * (0.5 / -a)) / (sqrt(fma((-4.0 * c), a, (b * b))) + b);
	} else {
		tmp = (-c - ((a * (c * c)) / (b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c))) * Float64(0.5 / Float64(-a))) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 85.0%

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

   if 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 96.8%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, c \cdot \left(-1 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.8%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. distribute-lft-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 90.5

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

   13. Applied rewrites 90.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{0.5}{-a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{-\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right)}{\left(2 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/
    (- (fma b b (fma (- b) b (* (* 4.0 a) c))))
    (* (* 2.0 a) (+ (sqrt (fma (* -4.0 c) a (* b b))) b)))
   (/ (- (- c) (/ (* a (* c c)) (* b b))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = -fma(b, b, fma(-b, b, ((4.0 * a) * c))) / ((2.0 * a) * (sqrt(fma((-4.0 * c), a, (b * b))) + b));
	} else {
		tmp = (-c - ((a * (c * c)) / (b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(-fma(b, b, fma(Float64(-b), b, Float64(Float64(4.0 * a) * c)))) / Float64(Float64(2.0 * a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[((-N[(b * b + N[((-b) * b + N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) / N[(N[(2.0 * a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   8. Applied rewrites 83.5%

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

   10. Applied rewrites 84.9%

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

   if 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 96.8%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, c \cdot \left(-1 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.8%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. distribute-lft-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 90.5

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

   13. Applied rewrites 90.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{-\mathsf{fma}\left(b, b, \mathsf{fma}\left(-b, b, \left(4 \cdot a\right) \cdot c\right)\right)}{\left(2 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 37.0)
     (/ (- (* b b) t_0) (* (* 2.0 a) (- (- b) (sqrt t_0))))
     (/ (- (- c) (/ (* a (* c c)) (* b b))) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 37.0) {
		tmp = ((b * b) - t_0) / ((2.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = (-c - ((a * (c * c)) / (b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(2.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 37.0], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   6. Applied rewrites 84.7%

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

   if 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 96.8%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, c \cdot \left(-1 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.8%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. distribute-lft-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 90.5

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

   13. Applied rewrites 90.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (/ (- (- c) (/ (* a (* c c)) (* b b))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = (-c - ((a * (c * c)) / (b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      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 83.4

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

   4. Applied rewrites 83.4%

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

   if 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 96.8%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, c \cdot \left(-1 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.8%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. distribute-lft-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 90.5

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

   13. Applied rewrites 90.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 37.0)
   (* (/ 0.5 a) (- (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/ (- (- c) (/ (* a (* c c)) (* b b))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 37.0) {
		tmp = (0.5 / a) * (sqrt(fma((-4.0 * c), a, (b * b))) - b);
	} else {
		tmp = (-c - ((a * (c * c)) / (b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 37.0)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 37.0], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 37

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      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 83.1

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

      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 83.1

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

   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 37 < b

   1. Initial program 43.3%

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 96.8%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, c \cdot \left(-1 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 96.8%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. distribute-lft-out N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 90.5

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

   13. Applied rewrites 90.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 37:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

4. Applied rewrites 54.2%

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

5. Taylor expanded in b around inf

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

7. Applied rewrites 92.1%

   \[\leadsto \frac{{a}^{-1}}{\color{blue}{b \cdot \left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, c \cdot \left(-1 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right)}} \]

8. Taylor expanded in a around 0

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

10. Applied rewrites 92.1%

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

11. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. distribute-lft-out N/A

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

    3. lower-*.f64 N/A

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

    4. lower-+.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 81.6

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

13. Applied rewrites 81.6%

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

14. Final simplification 81.6%

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

Alternative 17: 64.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 64.7

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

5. Applied rewrites 64.7%

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

Alternative 18: 1.6% accurate, 4.2× 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(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 54.2%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 64.7

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

5. Applied rewrites 64.7%

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

7. Applied rewrites 64.7%

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

9. Applied rewrites 64.7%

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

11. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2024319 
(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)))