Quadratic roots, narrow range

Percentage Accurate: 55.2% → 92.0%

Time: 9.1s

Alternatives: 20

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 12 \cdot \left(a \cdot a\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_2 := \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{t\_1}, b, t\_1\right)\right) \cdot -2\\ \mathbf{if}\;b \leq 0.258:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{t\_1}^{1.5}\right) \cdot {a}^{-1}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(6, \frac{t\_0}{{b}^{4}}, \frac{\left(a \cdot a\right) \cdot -64}{{b}^{4}}\right), \frac{\left(\mathsf{fma}\left(0.25, {t\_0}^{2}, \left(\mathsf{fma}\left(6, t\_0 \cdot a, {a}^{3} \cdot -64\right) \cdot a\right) \cdot -6\right) \cdot c\right) \cdot 0.5}{{b}^{6} \cdot a}\right), \left(\frac{a}{b \cdot b} \cdot 12\right) \cdot -0.5\right), 6\right) \cdot c\right) \cdot b}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 12.0 (* a a)))
        (t_1 (fma (* -4.0 a) c (* b b)))
        (t_2 (* (fma b b (fma (sqrt t_1) b t_1)) -2.0)))
   (if (<= b 0.258)
     (/ (* (fma (* b b) b (- (pow t_1 1.5))) (pow a -1.0)) t_2)
     (/
      (*
       (*
        (fma
         c
         (fma
          c
          (fma
           -0.5
           (fma 6.0 (/ t_0 (pow b 4.0)) (/ (* (* a a) -64.0) (pow b 4.0)))
           (/
            (*
             (*
              (fma
               0.25
               (pow t_0 2.0)
               (* (* (fma 6.0 (* t_0 a) (* (pow a 3.0) -64.0)) a) -6.0))
              c)
             0.5)
            (* (pow b 6.0) a)))
          (* (* (/ a (* b b)) 12.0) -0.5))
         6.0)
        c)
       b)
      t_2))))

C

double code(double a, double b, double c) {
	double t_0 = 12.0 * (a * a);
	double t_1 = fma((-4.0 * a), c, (b * b));
	double t_2 = fma(b, b, fma(sqrt(t_1), b, t_1)) * -2.0;
	double tmp;
	if (b <= 0.258) {
		tmp = (fma((b * b), b, -pow(t_1, 1.5)) * pow(a, -1.0)) / t_2;
	} else {
		tmp = ((fma(c, fma(c, fma(-0.5, fma(6.0, (t_0 / pow(b, 4.0)), (((a * a) * -64.0) / pow(b, 4.0))), (((fma(0.25, pow(t_0, 2.0), ((fma(6.0, (t_0 * a), (pow(a, 3.0) * -64.0)) * a) * -6.0)) * c) * 0.5) / (pow(b, 6.0) * a))), (((a / (b * b)) * 12.0) * -0.5)), 6.0) * c) * b) / t_2;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(12.0 * Float64(a * a))
	t_1 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_2 = Float64(fma(b, b, fma(sqrt(t_1), b, t_1)) * -2.0)
	tmp = 0.0
	if (b <= 0.258)
		tmp = Float64(Float64(fma(Float64(b * b), b, Float64(-(t_1 ^ 1.5))) * (a ^ -1.0)) / t_2);
	else
		tmp = Float64(Float64(Float64(fma(c, fma(c, fma(-0.5, fma(6.0, Float64(t_0 / (b ^ 4.0)), Float64(Float64(Float64(a * a) * -64.0) / (b ^ 4.0))), Float64(Float64(Float64(fma(0.25, (t_0 ^ 2.0), Float64(Float64(fma(6.0, Float64(t_0 * a), Float64((a ^ 3.0) * -64.0)) * a) * -6.0)) * c) * 0.5) / Float64((b ^ 6.0) * a))), Float64(Float64(Float64(a / Float64(b * b)) * 12.0) * -0.5)), 6.0) * c) * b) / t_2);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(12.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b + t$95$1), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, If[LessEqual[b, 0.258], N[(N[(N[(N[(b * b), $MachinePrecision] * b + (-N[Power[t$95$1, 1.5], $MachinePrecision])), $MachinePrecision] * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(c * N[(c * N[(-0.5 * N[(6.0 * N[(t$95$0 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -64.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[(N[(6.0 * N[(t$95$0 * a), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * -64.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * 0.5), $MachinePrecision] / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * 12.0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + 6.0), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.25800000000000001

   1. Initial program 89.5%

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

   4. Applied rewrites 89.7%

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

   6. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 90.9

         \[\leadsto \frac{{a}^{-1} \cdot \mathsf{fma}\left(b \cdot b, b, \color{blue}{-{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   8. Applied rewrites 90.9%

      \[\leadsto \frac{{a}^{-1} \cdot \color{blue}{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   if 0.25800000000000001 < b

   1. Initial program 51.4%

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

   4. Applied rewrites 51.4%

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

   6. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]

   7. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)}{a} + \left(\frac{-1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{a \cdot {b}^{4}} + \left(\frac{-1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}} + \frac{1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   9. Applied rewrites 92.1%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\left(a \cdot c\right) \cdot -12}{a}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-64 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}, 0.5 \cdot \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}, \left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right) \cdot \mathsf{fma}\left(-64 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{6}}\right)\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   10. Taylor expanded in c around 0

       \[\leadsto \frac{b \cdot \left(c \cdot \color{blue}{\left(6 + c \cdot \left(\frac{-1}{2} \cdot \left(-36 \cdot \frac{a}{{b}^{2}} + 48 \cdot \frac{a}{{b}^{2}}\right) + c \cdot \left(\frac{-1}{2} \cdot \left(-64 \cdot \frac{{a}^{2}}{{b}^{4}} + 6 \cdot \frac{-36 \cdot {a}^{2} + 48 \cdot {a}^{2}}{{b}^{4}}\right) + \frac{1}{2} \cdot \frac{c \cdot \left(-6 \cdot \left(a \cdot \left(-64 \cdot {a}^{3} + 6 \cdot \left(a \cdot \left(-36 \cdot {a}^{2} + 48 \cdot {a}^{2}\right)\right)\right)\right) + \frac{1}{4} \cdot {\left(-36 \cdot {a}^{2} + 48 \cdot {a}^{2}\right)}^{2}\right)}{a \cdot {b}^{6}}\right)\right)\right)}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   12. Applied rewrites 92.3%

       \[\leadsto \frac{b \cdot \left(c \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(6, \frac{\left(a \cdot a\right) \cdot 12}{{b}^{4}}, \frac{-64 \cdot \left(a \cdot a\right)}{{b}^{4}}\right), \frac{0.5 \cdot \left(c \cdot \mathsf{fma}\left(0.25, {\left(\left(a \cdot a\right) \cdot 12\right)}^{2}, -6 \cdot \left(a \cdot \mathsf{fma}\left(6, a \cdot \left(\left(a \cdot a\right) \cdot 12\right), -64 \cdot {a}^{3}\right)\right)\right)\right)}{a \cdot {b}^{6}}\right), -0.5 \cdot \left(\frac{a}{b \cdot b} \cdot 12\right)\right), 6\right)}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.258:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5, \mathsf{fma}\left(6, \frac{12 \cdot \left(a \cdot a\right)}{{b}^{4}}, \frac{\left(a \cdot a\right) \cdot -64}{{b}^{4}}\right), \frac{\left(\mathsf{fma}\left(0.25, {\left(12 \cdot \left(a \cdot a\right)\right)}^{2}, \left(\mathsf{fma}\left(6, \left(12 \cdot \left(a \cdot a\right)\right) \cdot a, {a}^{3} \cdot -64\right) \cdot a\right) \cdot -6\right) \cdot c\right) \cdot 0.5}{{b}^{6} \cdot a}\right), \left(\frac{a}{b \cdot b} \cdot 12\right) \cdot -0.5\right), 6\right) \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.258:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{t\_0}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{t\_0}, b, t\_0\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, \left(\left(-a\right) \cdot c\right) \cdot c, \mathsf{fma}\left(-0.25, \frac{20}{c \cdot c} \cdot \frac{{c}^{4} \cdot {a}^{4}}{a \cdot a}, \left(\left(a \cdot a\right) \cdot 2\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{-a}{{b}^{4}}, \frac{1}{a}\right)}{c}\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 0.258)
     (/
      (* (fma (* b b) b (- (pow t_0 1.5))) (pow a -1.0))
      (* (fma b b (fma (sqrt t_0) b t_0)) -2.0))
     (/
      (pow a -1.0)
      (*
       (-
        (-
         (/ 1.0 (* b b))
         (/
          (fma
           (- a)
           (* (* (- a) c) c)
           (fma
            -0.25
            (* (/ 20.0 (* c c)) (/ (* (pow c 4.0) (pow a 4.0)) (* a a)))
            (* (* (* a a) 2.0) (* c c))))
          (pow b 6.0)))
        (/ (fma (* c c) (/ (- a) (pow b 4.0)) (/ 1.0 a)) c))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 0.258) {
		tmp = (fma((b * b), b, -pow(t_0, 1.5)) * pow(a, -1.0)) / (fma(b, b, fma(sqrt(t_0), b, t_0)) * -2.0);
	} else {
		tmp = pow(a, -1.0) / ((((1.0 / (b * b)) - (fma(-a, ((-a * c) * c), fma(-0.25, ((20.0 / (c * c)) * ((pow(c, 4.0) * pow(a, 4.0)) / (a * a))), (((a * a) * 2.0) * (c * c)))) / pow(b, 6.0))) - (fma((c * c), (-a / pow(b, 4.0)), (1.0 / a)) / c)) * 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 <= 0.258)
		tmp = Float64(Float64(fma(Float64(b * b), b, Float64(-(t_0 ^ 1.5))) * (a ^ -1.0)) / Float64(fma(b, b, fma(sqrt(t_0), b, t_0)) * -2.0));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(1.0 / Float64(b * b)) - Float64(fma(Float64(-a), Float64(Float64(Float64(-a) * c) * c), fma(-0.25, Float64(Float64(20.0 / Float64(c * c)) * Float64(Float64((c ^ 4.0) * (a ^ 4.0)) / Float64(a * a))), Float64(Float64(Float64(a * a) * 2.0) * Float64(c * c)))) / (b ^ 6.0))) - Float64(fma(Float64(c * c), Float64(Float64(-a) / (b ^ 4.0)), Float64(1.0 / a)) / c)) * 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, 0.258], N[(N[(N[(N[(b * b), $MachinePrecision] * b + (-N[Power[t$95$0, 1.5], $MachinePrecision])), $MachinePrecision] * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * b + t$95$0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[((-a) * N[(N[((-a) * c), $MachinePrecision] * c), $MachinePrecision] + N[(-0.25 * N[(N[(20.0 / N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] * N[((-a) / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.25800000000000001

   1. Initial program 89.5%

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

   4. Applied rewrites 89.7%

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

   6. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 90.9

         \[\leadsto \frac{{a}^{-1} \cdot \mathsf{fma}\left(b \cdot b, b, \color{blue}{-{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   8. Applied rewrites 90.9%

      \[\leadsto \frac{{a}^{-1} \cdot \color{blue}{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   if 0.25800000000000001 < b

   1. Initial program 51.4%

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

   4. Applied rewrites 51.4%

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

   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 92.0%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 92.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.258:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-a, \left(\left(-a\right) \cdot c\right) \cdot c, \mathsf{fma}\left(-0.25, \frac{20}{c \cdot c} \cdot \frac{{c}^{4} \cdot {a}^{4}}{a \cdot a}, \left(\left(a \cdot a\right) \cdot 2\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{-a}{{b}^{4}}, \frac{1}{a}\right)}{c}\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{4}}\\ t_1 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.258:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{t\_1}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{t\_1}, b, t\_1\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(a \cdot a, -3, a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, t\_0 \cdot a, \mathsf{fma}\left(a, t\_0, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 4.0))) (t_1 (fma (* -4.0 a) c (* b b))))
   (if (<= b 0.258)
     (/
      (* (fma (* b b) b (- (pow t_1 1.5))) (pow a -1.0))
      (* (fma b b (fma (sqrt t_1) b t_1)) -2.0))
     (/
      (pow a -1.0)
      (*
       (-
        (-
         (/ 1.0 (* b b))
         (/ (* (fma (* a a) -3.0 (* a a)) (* c c)) (pow b 6.0)))
        (fma -2.0 (* t_0 a) (fma a t_0 (/ (/ 1.0 a) c))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 4.0);
	double t_1 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 0.258) {
		tmp = (fma((b * b), b, -pow(t_1, 1.5)) * pow(a, -1.0)) / (fma(b, b, fma(sqrt(t_1), b, t_1)) * -2.0);
	} else {
		tmp = pow(a, -1.0) / ((((1.0 / (b * b)) - ((fma((a * a), -3.0, (a * a)) * (c * c)) / pow(b, 6.0))) - fma(-2.0, (t_0 * a), fma(a, t_0, ((1.0 / a) / c)))) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 4.0))
	t_1 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 0.258)
		tmp = Float64(Float64(fma(Float64(b * b), b, Float64(-(t_1 ^ 1.5))) * (a ^ -1.0)) / Float64(fma(b, b, fma(sqrt(t_1), b, t_1)) * -2.0));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(1.0 / Float64(b * b)) - Float64(Float64(fma(Float64(a * a), -3.0, Float64(a * a)) * Float64(c * c)) / (b ^ 6.0))) - fma(-2.0, Float64(t_0 * a), fma(a, t_0, Float64(Float64(1.0 / a) / c)))) * b));
	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 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.258], N[(N[(N[(N[(b * b), $MachinePrecision] * b + (-N[Power[t$95$1, 1.5], $MachinePrecision])), $MachinePrecision] * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b + t$95$1), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(a * a), $MachinePrecision] * -3.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(t$95$0 * a), $MachinePrecision] + N[(a * t$95$0 + N[(N[(1.0 / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.25800000000000001

   1. Initial program 89.5%

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

   4. Applied rewrites 89.7%

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

   6. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 90.9

         \[\leadsto \frac{{a}^{-1} \cdot \mathsf{fma}\left(b \cdot b, b, \color{blue}{-{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   8. Applied rewrites 90.9%

      \[\leadsto \frac{{a}^{-1} \cdot \color{blue}{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   if 0.25800000000000001 < b

   1. Initial program 51.4%

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

   4. Applied rewrites 51.4%

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

   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 92.0%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 92.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.258:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(a \cdot a, -3, a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}}\right) - \mathsf{fma}\left(-2, \frac{c}{{b}^{4}} \cdot a, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.258:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{t\_0}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{t\_0}, b, t\_0\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot a, {c}^{4}, {c}^{3} \cdot \left(-2 \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}}, a, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 0.258)
     (/
      (* (fma (* b b) b (- (pow t_0 1.5))) (pow a -1.0))
      (* (fma b b (fma (sqrt t_0) b t_0)) -2.0))
     (fma
      (fma
       (/
        (fma (* -5.0 a) (pow c 4.0) (* (pow c 3.0) (* -2.0 (* b b))))
        (pow b 7.0))
       a
       (* (/ c (pow b 3.0)) (- c)))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 0.258) {
		tmp = (fma((b * b), b, -pow(t_0, 1.5)) * pow(a, -1.0)) / (fma(b, b, fma(sqrt(t_0), b, t_0)) * -2.0);
	} else {
		tmp = fma(fma((fma((-5.0 * a), pow(c, 4.0), (pow(c, 3.0) * (-2.0 * (b * b)))) / pow(b, 7.0)), a, ((c / pow(b, 3.0)) * -c)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 0.258)
		tmp = Float64(Float64(fma(Float64(b * b), b, Float64(-(t_0 ^ 1.5))) * (a ^ -1.0)) / Float64(fma(b, b, fma(sqrt(t_0), b, t_0)) * -2.0));
	else
		tmp = fma(fma(Float64(fma(Float64(-5.0 * a), (c ^ 4.0), Float64((c ^ 3.0) * Float64(-2.0 * Float64(b * b)))) / (b ^ 7.0)), a, Float64(Float64(c / (b ^ 3.0)) * Float64(-c))), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.258], N[(N[(N[(N[(b * b), $MachinePrecision] * b + (-N[Power[t$95$0, 1.5], $MachinePrecision])), $MachinePrecision] * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * b + t$95$0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-5.0 * a), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[Power[c, 3.0], $MachinePrecision] * N[(-2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.25800000000000001

   1. Initial program 89.5%

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

   4. Applied rewrites 89.7%

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

   6. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 90.9

         \[\leadsto \frac{{a}^{-1} \cdot \mathsf{fma}\left(b \cdot b, b, \color{blue}{-{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   8. Applied rewrites 90.9%

      \[\leadsto \frac{{a}^{-1} \cdot \color{blue}{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   if 0.25800000000000001 < b

   1. Initial program 51.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 91.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.258:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot a, {c}^{4}, {c}^{3} \cdot \left(-2 \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}}, a, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{t\_0}, b, t\_0\right)\right) \cdot -2\\ \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{t\_0}^{1.5}\right) \cdot {a}^{-1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(6, \frac{12 \cdot \left(a \cdot a\right)}{{b}^{4}}, \frac{\left(a \cdot a\right) \cdot -64}{{b}^{4}}\right), \frac{a}{b \cdot b} \cdot 12\right) \cdot -0.5, 6\right) \cdot c\right) \cdot b}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b)))
        (t_1 (* (fma b b (fma (sqrt t_0) b t_0)) -2.0)))
   (if (<= b 2.8)
     (/ (* (fma (* b b) b (- (pow t_0 1.5))) (pow a -1.0)) t_1)
     (/
      (*
       (*
        (fma
         c
         (*
          (fma
           c
           (fma
            6.0
            (/ (* 12.0 (* a a)) (pow b 4.0))
            (/ (* (* a a) -64.0) (pow b 4.0)))
           (* (/ a (* b b)) 12.0))
          -0.5)
         6.0)
        c)
       b)
      t_1))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double t_1 = fma(b, b, fma(sqrt(t_0), b, t_0)) * -2.0;
	double tmp;
	if (b <= 2.8) {
		tmp = (fma((b * b), b, -pow(t_0, 1.5)) * pow(a, -1.0)) / t_1;
	} else {
		tmp = ((fma(c, (fma(c, fma(6.0, ((12.0 * (a * a)) / pow(b, 4.0)), (((a * a) * -64.0) / pow(b, 4.0))), ((a / (b * b)) * 12.0)) * -0.5), 6.0) * c) * b) / t_1;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_1 = Float64(fma(b, b, fma(sqrt(t_0), b, t_0)) * -2.0)
	tmp = 0.0
	if (b <= 2.8)
		tmp = Float64(Float64(fma(Float64(b * b), b, Float64(-(t_0 ^ 1.5))) * (a ^ -1.0)) / t_1);
	else
		tmp = Float64(Float64(Float64(fma(c, Float64(fma(c, fma(6.0, Float64(Float64(12.0 * Float64(a * a)) / (b ^ 4.0)), Float64(Float64(Float64(a * a) * -64.0) / (b ^ 4.0))), Float64(Float64(a / Float64(b * b)) * 12.0)) * -0.5), 6.0) * c) * b) / t_1);
	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]}, Block[{t$95$1 = N[(N[(b * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * b + t$95$0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, If[LessEqual[b, 2.8], N[(N[(N[(N[(b * b), $MachinePrecision] * b + (-N[Power[t$95$0, 1.5], $MachinePrecision])), $MachinePrecision] * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(c * N[(N[(c * N[(6.0 * N[(N[(12.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -64.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + 6.0), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999998

   1. Initial program 85.3%

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

   4. Applied rewrites 85.3%

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

   6. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 86.3

         \[\leadsto \frac{{a}^{-1} \cdot \mathsf{fma}\left(b \cdot b, b, \color{blue}{-{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   8. Applied rewrites 86.3%

      \[\leadsto \frac{{a}^{-1} \cdot \color{blue}{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   if 2.7999999999999998 < b

   1. Initial program 49.0%

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

   4. Applied rewrites 49.0%

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

   6. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]

   7. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)}{a} + \left(\frac{-1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{a \cdot {b}^{4}} + \left(\frac{-1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}} + \frac{1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   9. Applied rewrites 93.0%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\left(a \cdot c\right) \cdot -12}{a}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-64 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}, 0.5 \cdot \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}, \left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right) \cdot \mathsf{fma}\left(-64 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{6}}\right)\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   10. Taylor expanded in c around 0

       \[\leadsto \frac{b \cdot \left(c \cdot \color{blue}{\left(6 + c \cdot \left(\frac{-1}{2} \cdot \left(c \cdot \left(-64 \cdot \frac{{a}^{2}}{{b}^{4}} + 6 \cdot \frac{-36 \cdot {a}^{2} + 48 \cdot {a}^{2}}{{b}^{4}}\right)\right) + \frac{-1}{2} \cdot \left(-36 \cdot \frac{a}{{b}^{2}} + 48 \cdot \frac{a}{{b}^{2}}\right)\right)\right)}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 90.7%

       \[\leadsto \frac{b \cdot \left(c \cdot \color{blue}{\mathsf{fma}\left(c, -0.5 \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(6, \frac{\left(a \cdot a\right) \cdot 12}{{b}^{4}}, \frac{-64 \cdot \left(a \cdot a\right)}{{b}^{4}}\right), \frac{a}{b \cdot b} \cdot 12\right), 6\right)}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(6, \frac{12 \cdot \left(a \cdot a\right)}{{b}^{4}}, \frac{\left(a \cdot a\right) \cdot -64}{{b}^{4}}\right), \frac{a}{b \cdot b} \cdot 12\right) \cdot -0.5, 6\right) \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{t\_0}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{t\_0}, b, t\_0\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\frac{\frac{-1}{a}}{c} - \left(\frac{-1}{b \cdot b} - \frac{c \cdot a}{{b}^{4}}\right)\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 2.8)
     (/
      (* (fma (* b b) b (- (pow t_0 1.5))) (pow a -1.0))
      (* (fma b b (fma (sqrt t_0) b t_0)) -2.0))
     (/
      (pow a -1.0)
      (*
       (- (/ (/ -1.0 a) c) (- (/ -1.0 (* b b)) (/ (* c a) (pow b 4.0))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 2.8) {
		tmp = (fma((b * b), b, -pow(t_0, 1.5)) * pow(a, -1.0)) / (fma(b, b, fma(sqrt(t_0), b, t_0)) * -2.0);
	} else {
		tmp = pow(a, -1.0) / ((((-1.0 / a) / c) - ((-1.0 / (b * b)) - ((c * a) / pow(b, 4.0)))) * 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 <= 2.8)
		tmp = Float64(Float64(fma(Float64(b * b), b, Float64(-(t_0 ^ 1.5))) * (a ^ -1.0)) / Float64(fma(b, b, fma(sqrt(t_0), b, t_0)) * -2.0));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(-1.0 / a) / c) - Float64(Float64(-1.0 / Float64(b * b)) - Float64(Float64(c * a) / (b ^ 4.0)))) * 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, 2.8], N[(N[(N[(N[(b * b), $MachinePrecision] * b + (-N[Power[t$95$0, 1.5], $MachinePrecision])), $MachinePrecision] * N[Power[a, -1.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * b + t$95$0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(-1.0 / a), $MachinePrecision] / c), $MachinePrecision] - N[(N[(-1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999998

   1. Initial program 85.3%

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

   4. Applied rewrites 85.3%

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

   6. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 86.3

         \[\leadsto \frac{{a}^{-1} \cdot \mathsf{fma}\left(b \cdot b, b, \color{blue}{-{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   8. Applied rewrites 86.3%

      \[\leadsto \frac{{a}^{-1} \cdot \color{blue}{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   if 2.7999999999999998 < b

   1. Initial program 49.0%

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 90.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, b, -{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right) \cdot {a}^{-1}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\frac{\frac{-1}{a}}{c} - \left(\frac{-1}{b \cdot b} - \frac{c \cdot a}{{b}^{4}}\right)\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;{\left(\left(-b\right) - \sqrt{t\_0}\right)}^{-1} \cdot {\left(\frac{2 \cdot a}{b \cdot b - t\_0}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\frac{\frac{-1}{a}}{c} - \left(\frac{-1}{b \cdot b} - \frac{c \cdot a}{{b}^{4}}\right)\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 2.8)
     (*
      (pow (- (- b) (sqrt t_0)) -1.0)
      (pow (/ (* 2.0 a) (- (* b b) t_0)) -1.0))
     (/
      (pow a -1.0)
      (*
       (- (/ (/ -1.0 a) c) (- (/ -1.0 (* b b)) (/ (* c a) (pow b 4.0))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 2.8) {
		tmp = pow((-b - sqrt(t_0)), -1.0) * pow(((2.0 * a) / ((b * b) - t_0)), -1.0);
	} else {
		tmp = pow(a, -1.0) / ((((-1.0 / a) / c) - ((-1.0 / (b * b)) - ((c * a) / pow(b, 4.0)))) * 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 <= 2.8)
		tmp = Float64((Float64(Float64(-b) - sqrt(t_0)) ^ -1.0) * (Float64(Float64(2.0 * a) / Float64(Float64(b * b) - t_0)) ^ -1.0));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(-1.0 / a) / c) - Float64(Float64(-1.0 / Float64(b * b)) - Float64(Float64(c * a) / (b ^ 4.0)))) * 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, 2.8], N[(N[Power[N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(2.0 * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(-1.0 / a), $MachinePrecision] / c), $MachinePrecision] - N[(N[(-1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999998

   1. Initial program 85.3%

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

   4. Applied rewrites 85.3%

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

   6. Applied rewrites 86.2%

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

   if 2.7999999999999998 < b

   1. Initial program 49.0%

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 90.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}^{-1} \cdot {\left(\frac{2 \cdot a}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\frac{\frac{-1}{a}}{c} - \left(\frac{-1}{b \cdot b} - \frac{c \cdot a}{{b}^{4}}\right)\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;{\left(\left(-b\right) - \sqrt{t\_0}\right)}^{-1} \cdot {\left(\frac{2 \cdot a}{b \cdot b - t\_0}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{a}{{b}^{3}}, \frac{1}{b}\right), \frac{-b}{a}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 2.8)
     (*
      (pow (- (- b) (sqrt t_0)) -1.0)
      (pow (/ (* 2.0 a) (- (* b b) t_0)) -1.0))
     (/
      (pow a -1.0)
      (/ (fma c (fma c (/ a (pow b 3.0)) (/ 1.0 b)) (/ (- b) a)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 2.8) {
		tmp = pow((-b - sqrt(t_0)), -1.0) * pow(((2.0 * a) / ((b * b) - t_0)), -1.0);
	} else {
		tmp = pow(a, -1.0) / (fma(c, fma(c, (a / pow(b, 3.0)), (1.0 / b)), (-b / a)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 2.8)
		tmp = Float64((Float64(Float64(-b) - sqrt(t_0)) ^ -1.0) * (Float64(Float64(2.0 * a) / Float64(Float64(b * b) - t_0)) ^ -1.0));
	else
		tmp = Float64((a ^ -1.0) / Float64(fma(c, fma(c, Float64(a / (b ^ 3.0)), Float64(1.0 / b)), 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 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.8], N[(N[Power[N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[(N[(2.0 * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(c * N[(c * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999998

   1. Initial program 85.3%

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

   4. Applied rewrites 85.3%

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

   6. Applied rewrites 86.2%

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

   if 2.7999999999999998 < b

   1. Initial program 49.0%

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 90.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}^{-1} \cdot {\left(\frac{2 \cdot a}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{a}{{b}^{3}}, \frac{1}{b}\right), \frac{-b}{a}\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{a}{{b}^{3}}, \frac{1}{b}\right), \frac{-b}{a}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 2.8)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (pow a -1.0)
      (/ (fma c (fma c (/ a (pow b 3.0)) (/ 1.0 b)) (/ (- b) a)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 2.8) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / (fma(c, fma(c, (a / pow(b, 3.0)), (1.0 / b)), (-b / a)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 2.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(fma(c, fma(c, Float64(a / (b ^ 3.0)), Float64(1.0 / b)), 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 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(c * N[(c * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999998

   1. Initial program 85.3%

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

   4. Applied rewrites 85.3%

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

   6. Applied rewrites 86.2%

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

   if 2.7999999999999998 < b

   1. Initial program 49.0%

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 90.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{a}{{b}^{3}}, \frac{1}{b}\right), \frac{-b}{a}\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1, c, \frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 2.8)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (fma
       -2.0
       (/ (* (* (* a a) c) (* c c)) (pow b 4.0))
       (fma -1.0 c (/ (* (* c c) a) (* (- 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 <= 2.8) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma(-2.0, ((((a * a) * c) * (c * c)) / pow(b, 4.0)), fma(-1.0, c, (((c * c) * a) / (-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 <= 2.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(fma(-2.0, Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) / (b ^ 4.0)), fma(-1.0, c, Float64(Float64(Float64(c * c) * a) / Float64(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, 2.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * c + N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999998

   1. Initial program 85.3%

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

   4. Applied rewrites 85.3%

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

   6. Applied rewrites 86.2%

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

   if 2.7999999999999998 < b

   1. Initial program 49.0%

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

   3. Taylor expanded in 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 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 90.2%

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

   12. Applied rewrites 90.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1, c, \frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{{b}^{4}} - \frac{a}{b \cdot b}, -1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 2.8)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (*
       (fma c (- (/ (* (* (* a a) c) -2.0) (pow b 4.0)) (/ a (* b b))) -1.0)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 2.8) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = (fma(c, (((((a * a) * c) * -2.0) / pow(b, 4.0)) - (a / (b * b))), -1.0) * c) / 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 <= 2.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(Float64(fma(c, Float64(Float64(Float64(Float64(Float64(a * a) * c) * -2.0) / (b ^ 4.0)) - Float64(a / Float64(b * b))), -1.0) * c) / 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, 2.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999998

   1. Initial program 85.3%

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

   4. Applied rewrites 85.3%

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

   6. Applied rewrites 86.2%

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

   if 2.7999999999999998 < b

   1. Initial program 49.0%

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

   3. Taylor expanded in 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 69.7

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

   5. Applied rewrites 69.7%

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

   7. Applied rewrites 69.6%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 90.2%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 90.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{{b}^{4}} - \frac{a}{b \cdot b}, -1\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{t\_1 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5, \left(\frac{a}{b \cdot b} \cdot 12\right) \cdot c, 6\right) \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(t\_1, b, t\_0\right)\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 4.8)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ t_1 b))
     (/
      (* (* (fma -0.5 (* (* (/ a (* b b)) 12.0) c) 6.0) c) b)
      (* (fma b b (fma t_1 b t_0)) -2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 4.8) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (t_1 + b);
	} else {
		tmp = ((fma(-0.5, (((a / (b * b)) * 12.0) * c), 6.0) * c) * b) / (fma(b, b, fma(t_1, b, t_0)) * -2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(t_1 + b));
	else
		tmp = Float64(Float64(Float64(fma(-0.5, Float64(Float64(Float64(a / Float64(b * b)) * 12.0) * c), 6.0) * c) * b) / Float64(fma(b, b, fma(t_1, b, t_0)) * -2.0));
	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]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * 12.0), $MachinePrecision] * c), $MachinePrecision] + 6.0), $MachinePrecision] * c), $MachinePrecision] * b), $MachinePrecision] / N[(N[(b * b + N[(t$95$1 * b + t$95$0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.0%

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

   4. Applied rewrites 85.0%

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

   6. Applied rewrites 85.8%

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

   if 4.79999999999999982 < b

   1. Initial program 48.3%

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

   4. Applied rewrites 48.3%

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

   6. Applied rewrites 49.5%

      \[\leadsto \color{blue}{\frac{{a}^{-1} \cdot \left({b}^{3} - {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)}} \]

   7. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)}{a} + \left(\frac{-1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{a \cdot {b}^{4}} + \left(\frac{-1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}} + \frac{1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   9. Applied rewrites 92.9%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\left(a \cdot c\right) \cdot -12}{a}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(-64 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}, 0.5 \cdot \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}, \left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right) \cdot \mathsf{fma}\left(-64 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{6}}\right)\right)}}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]

   10. Taylor expanded in c around 0

       \[\leadsto \frac{b \cdot \left(c \cdot \color{blue}{\left(6 + \frac{-1}{2} \cdot \left(c \cdot \left(-36 \cdot \frac{a}{{b}^{2}} + 48 \cdot \frac{a}{{b}^{2}}\right)\right)\right)}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 86.1%

       \[\leadsto \frac{b \cdot \left(c \cdot \color{blue}{\mathsf{fma}\left(-0.5, c \cdot \left(\frac{a}{b \cdot b} \cdot 12\right), 6\right)}\right)}{-2 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5, \left(\frac{a}{b \cdot b} \cdot 12\right) \cdot c, 6\right) \cdot c\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)\right) \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 4.8)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/ (fma a (/ (* c c) (* b b)) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 4.8) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -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 <= 4.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-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, 4.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.0%

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

   4. Applied rewrites 85.0%

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

   6. Applied rewrites 85.8%

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

   if 4.79999999999999982 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in 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 70.2

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 1.6%

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

   10. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.6

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

   12. Applied rewrites 85.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 4.8)
     (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) (* 2.0 a)))
     (/ (fma a (/ (* c c) (* b b)) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 4.8) {
		tmp = (t_0 - (b * b)) / ((sqrt(t_0) + b) * (2.0 * a));
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -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 <= 4.8)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * Float64(2.0 * a)));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-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, 4.8], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.0%

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

   4. Applied rewrites 85.0%

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

   6. Applied rewrites 85.7%

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

   if 4.79999999999999982 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in 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 70.2

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 1.6%

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

   10. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.6

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

   12. Applied rewrites 85.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{1}{\frac{2}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8)
   (/ 1.0 (* (/ 2.0 (- (sqrt (fma (* -4.0 a) c (* b b))) b)) a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = 1.0 / ((2.0 / (sqrt(fma((-4.0 * a), c, (b * b))) - b)) * a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(1.0 / Float64(Float64(2.0 / Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b)) * a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8], N[(1.0 / N[(N[(2.0 / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.0%

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

   4. Applied rewrites 85.0%

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

   6. Applied rewrites 85.1%

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

   if 4.79999999999999982 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in 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 70.2

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 1.6%

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

   10. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.6

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

   12. Applied rewrites 85.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{1}{\frac{2}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8)
   (/ (- (sqrt (fma b b (* (* c -4.0) a))) b) (* 2.0 a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = (sqrt(fma(b, b, ((c * -4.0) * a))) - b) / (2.0 * a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -4.0) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 85.0

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

   4. Applied rewrites 85.0%

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

   if 4.79999999999999982 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in 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 70.2

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 1.6%

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

   10. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.6

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

   12. Applied rewrites 85.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8)
   (* (- (sqrt (fma (* c -4.0) a (* b b))) b) (/ 0.5 a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 85.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 85.0

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

   4. Applied rewrites 85.0%

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

   if 4.79999999999999982 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in 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 70.2

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

   5. Applied rewrites 70.2%

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

   7. Applied rewrites 70.1%

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

   9. Applied rewrites 1.6%

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

   10. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.6

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

   12. Applied rewrites 85.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 81.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in 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 63.2

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

5. Applied rewrites 63.2%

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

7. Applied rewrites 63.1%

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

9. Applied rewrites 1.6%

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

10. Taylor expanded in b around inf

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

    1. distribute-lft-out N/A

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

    2. associate-*r/ N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. +-commutative N/A

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

    7. associate-/l* N/A

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

    8. lower-fma.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f64 78.9

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

12. Applied rewrites 78.9%

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

13. Final simplification 78.9%

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

Alternative 19: 64.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in 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 63.2

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

5. Applied rewrites 63.2%

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

Alternative 20: 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 56.3%

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

3. Taylor expanded in 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 63.2

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

5. Applied rewrites 63.2%

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

7. Applied rewrites 63.1%

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

9. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2024332 
(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)))