Quadratic roots, narrow range

Percentage Accurate: 55.0% → 92.3%

Time: 11.7s

Alternatives: 18

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.0% 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: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - t\_0} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\mathsf{fma}\left(\left(-c\right) \cdot c, \frac{\left(a \cdot a\right) \cdot -2}{{b}^{6}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, {\left(b \cdot b\right)}^{-1} - \frac{{a}^{-1}}{c}\right)\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.16)
     (/ (pow a -1.0) (* (/ -2.0 (- (* b b) t_0)) (+ (sqrt t_0) b)))
     (/
      (pow a -1.0)
      (*
       (fma
        (* (- c) c)
        (/ (* (* a a) -2.0) (pow b 6.0))
        (fma a (/ c (pow b 4.0)) (- (pow (* b b) -1.0) (/ (pow a -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 <= 1.16) {
		tmp = pow(a, -1.0) / ((-2.0 / ((b * b) - t_0)) * (sqrt(t_0) + b));
	} else {
		tmp = pow(a, -1.0) / (fma((-c * c), (((a * a) * -2.0) / pow(b, 6.0)), fma(a, (c / pow(b, 4.0)), (pow((b * b), -1.0) - (pow(a, -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 <= 1.16)
		tmp = Float64((a ^ -1.0) / Float64(Float64(-2.0 / Float64(Float64(b * b) - t_0)) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64((a ^ -1.0) / Float64(fma(Float64(Float64(-c) * c), Float64(Float64(Float64(a * a) * -2.0) / (b ^ 6.0)), fma(a, Float64(c / (b ^ 4.0)), Float64((Float64(b * b) ^ -1.0) - Float64((a ^ -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, 1.16], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(-2.0 / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[((-c) * c), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(b * b), $MachinePrecision], -1.0], $MachinePrecision] - N[(N[Power[a, -1.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15999999999999992

   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 85.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 85.2

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

   6. Applied rewrites 85.2%

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

   8. Applied rewrites 86.6%

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

   if 1.15999999999999992 < b

   1. Initial program 50.0%

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.7%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{\frac{-a}{{b}^{3}}}{b \cdot b}, \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right) \cdot b}{a}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right), -c, \frac{a}{{b}^{3}}\right), c, \frac{1}{b}\right), c, \frac{-b}{a}\right)}{c}}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.7%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 93.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\mathsf{fma}\left(\left(-c\right) \cdot c, \frac{\left(a \cdot a\right) \cdot -2}{{b}^{6}}, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, {\left(b \cdot b\right)}^{-1} - \frac{{a}^{-1}}{c}\right)\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - t\_0} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot -2}{{b}^{5}}, -c, \frac{a}{{b}^{3}}\right), c, {b}^{-1}\right), c, \frac{-b}{a}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.16)
     (/ (pow a -1.0) (* (/ -2.0 (- (* b b) t_0)) (+ (sqrt t_0) b)))
     (/
      (pow a -1.0)
      (/
       (fma
        (fma
         (fma (/ (* (* a a) -2.0) (pow b 5.0)) (- c) (/ a (pow b 3.0)))
         c
         (pow b -1.0))
        c
        (/ (- b) a))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 1.16) {
		tmp = pow(a, -1.0) / ((-2.0 / ((b * b) - t_0)) * (sqrt(t_0) + b));
	} else {
		tmp = pow(a, -1.0) / (fma(fma(fma((((a * a) * -2.0) / pow(b, 5.0)), -c, (a / pow(b, 3.0))), c, pow(b, -1.0)), c, (-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 <= 1.16)
		tmp = Float64((a ^ -1.0) / Float64(Float64(-2.0 / Float64(Float64(b * b) - t_0)) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64((a ^ -1.0) / Float64(fma(fma(fma(Float64(Float64(Float64(a * a) * -2.0) / (b ^ 5.0)), Float64(-c), Float64(a / (b ^ 3.0))), c, (b ^ -1.0)), c, Float64(Float64(-b) / a)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.16], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(-2.0 / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * (-c) + N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * c + N[((-b) / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15999999999999992

   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 85.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 85.2

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

   6. Applied rewrites 85.2%

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

   8. Applied rewrites 86.6%

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

   if 1.15999999999999992 < b

   1. Initial program 50.0%

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.7%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{\frac{-a}{{b}^{3}}}{b \cdot b}, \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right) \cdot b}{a}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right), -c, \frac{a}{{b}^{3}}\right), c, \frac{1}{b}\right), c, \frac{-b}{a}\right)}{c}}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.7%

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

       1. lift-pow.f64 N/A

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

       2. inv-pow N/A

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

       3. lift-/.f64 93.7

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

   12. Applied rewrites 93.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot -2}{{b}^{5}}, -c, \frac{a}{{b}^{3}}\right), c, {b}^{-1}\right), c, \frac{-b}{a}\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - t\_0} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\left(-c\right) \cdot c, \left(a \cdot a\right) \cdot -2, \mathsf{fma}\left(\left(1 - \frac{\frac{b \cdot b}{a}}{c}\right) \cdot b, b, c \cdot a\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.16)
     (/ (pow a -1.0) (* (/ -2.0 (- (* b b) t_0)) (+ (sqrt t_0) b)))
     (/
      (pow a -1.0)
      (/
       (fma
        (* (- c) c)
        (* (* a a) -2.0)
        (* (fma (* (- 1.0 (/ (/ (* b b) a) c)) b) b (* c a)) (* b b)))
       (pow b 5.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 1.16) {
		tmp = pow(a, -1.0) / ((-2.0 / ((b * b) - t_0)) * (sqrt(t_0) + b));
	} else {
		tmp = pow(a, -1.0) / (fma((-c * c), ((a * a) * -2.0), (fma(((1.0 - (((b * b) / a) / c)) * b), b, (c * a)) * (b * b))) / pow(b, 5.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 1.16)
		tmp = Float64((a ^ -1.0) / Float64(Float64(-2.0 / Float64(Float64(b * b) - t_0)) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64((a ^ -1.0) / Float64(fma(Float64(Float64(-c) * c), Float64(Float64(a * a) * -2.0), Float64(fma(Float64(Float64(1.0 - Float64(Float64(Float64(b * b) / a) / c)) * b), b, Float64(c * a)) * Float64(b * b))) / (b ^ 5.0)));
	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]}, If[LessEqual[b, 1.16], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(-2.0 / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[((-c) * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(N[(N[(1.0 - N[(N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(c * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15999999999999992

   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 85.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 85.2

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

   6. Applied rewrites 85.2%

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

   8. Applied rewrites 86.6%

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

   if 1.15999999999999992 < b

   1. Initial program 50.0%

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.7%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{\frac{-a}{{b}^{3}}}{b \cdot b}, \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right) \cdot b}{a}, \frac{2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right)\right), -c, \frac{a}{{b}^{3}}\right), c, \frac{1}{b}\right), c, \frac{-b}{a}\right)}{c}}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.7%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 93.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\left(-c\right) \cdot c, \left(a \cdot a\right) \cdot -2, \mathsf{fma}\left(\left(1 - \frac{\frac{b \cdot b}{a}}{c}\right) \cdot b, b, c \cdot a\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - t\_0} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(b, \frac{-1}{a}, \frac{c}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -5.2e-5)
     (/ (pow a -1.0) (* (/ -2.0 (- (* b b) t_0)) (+ (sqrt t_0) b)))
     (/ (pow a -1.0) (/ (fma b (/ -1.0 a) (/ c b)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -5.2e-5) {
		tmp = pow(a, -1.0) / ((-2.0 / ((b * b) - t_0)) * (sqrt(t_0) + b));
	} else {
		tmp = pow(a, -1.0) / (fma(b, (-1.0 / a), (c / b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -5.2e-5)
		tmp = Float64((a ^ -1.0) / Float64(Float64(-2.0 / Float64(Float64(b * b) - t_0)) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64((a ^ -1.0) / Float64(fma(b, Float64(-1.0 / a), Float64(c / b)) / 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[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], -5.2e-5], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(-2.0 / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(b * N[(-1.0 / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.19999999999999968e-5

   1. Initial program 78.1%

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

   4. Applied rewrites 78.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 78.1

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

   6. Applied rewrites 78.1%

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

   8. Applied rewrites 79.7%

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

   if -5.19999999999999968e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 35.9%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.6

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

   7. Applied rewrites 94.6%

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

   9. Applied rewrites 94.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(b, \frac{-1}{a}, \frac{c}{b}\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - t\_0} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{{b}^{3}}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -5.2e-5)
     (/ (pow a -1.0) (* (/ -2.0 (- (* b b) t_0)) (+ (sqrt t_0) b)))
     (- (fma a (/ (* c c) (pow b 3.0)) (/ 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 + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -5.2e-5) {
		tmp = pow(a, -1.0) / ((-2.0 / ((b * b) - t_0)) * (sqrt(t_0) + b));
	} else {
		tmp = -fma(a, ((c * c) / pow(b, 3.0)), (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 (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -5.2e-5)
		tmp = Float64((a ^ -1.0) / Float64(Float64(-2.0 / Float64(Float64(b * b) - t_0)) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / (b ^ 3.0)), 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[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], -5.2e-5], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(-2.0 / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.19999999999999968e-5

   1. Initial program 78.1%

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

   4. Applied rewrites 78.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 78.1

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

   6. Applied rewrites 78.1%

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

   8. Applied rewrites 79.7%

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

   if -5.19999999999999968e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 35.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 94.7

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

   8. Applied rewrites 94.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{{b}^{3}}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - t\_0} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -5.2e-5)
     (/ (pow a -1.0) (* (/ -2.0 (- (* b b) t_0)) (+ (sqrt t_0) b)))
     (/ (pow a -1.0) (/ (- (/ c 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 + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -5.2e-5) {
		tmp = pow(a, -1.0) / ((-2.0 / ((b * b) - t_0)) * (sqrt(t_0) + b));
	} else {
		tmp = pow(a, -1.0) / (((c / 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 (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -5.2e-5)
		tmp = Float64((a ^ -1.0) / Float64(Float64(-2.0 / Float64(Float64(b * b) - t_0)) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(c / b) - 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[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], -5.2e-5], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(-2.0 / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.19999999999999968e-5

   1. Initial program 78.1%

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

   4. Applied rewrites 78.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 78.1

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

   6. Applied rewrites 78.1%

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

   8. Applied rewrites 79.7%

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

   if -5.19999999999999968e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 35.9%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.6

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

   7. Applied rewrites 94.6%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 94.6

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

   9. Applied rewrites 94.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c a) -4.0 (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -5.2e-5)
     (/ (* (- (* b b) t_0) (/ 0.5 a)) (- (- b) (sqrt t_0)))
     (/ (pow a -1.0) (/ (- (/ c b) (/ b a)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * a), -4.0, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -5.2e-5) {
		tmp = (((b * b) - t_0) * (0.5 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = pow(a, -1.0) / (((c / b) - (b / a)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * a), -4.0, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -5.2e-5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(0.5 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(c / b) - Float64(b / a)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -5.2e-5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.19999999999999968e-5

   1. Initial program 78.1%

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 79.7%

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

   if -5.19999999999999968e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 35.9%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.6

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

   7. Applied rewrites 94.6%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 94.6

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

   9. Applied rewrites 94.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(b \cdot b - \mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c a) -4.0 (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -5.2e-5)
     (/ (- (* b b) t_0) (* (* 2.0 a) (- (- b) (sqrt t_0))))
     (/ (pow a -1.0) (/ (- (/ c b) (/ b a)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * a), -4.0, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -5.2e-5) {
		tmp = ((b * b) - t_0) / ((2.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = pow(a, -1.0) / (((c / b) - (b / a)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * a), -4.0, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -5.2e-5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(2.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(c / b) - Float64(b / a)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -5.2e-5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.19999999999999968e-5

   1. Initial program 78.1%

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 79.7%

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

   if -5.19999999999999968e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 35.9%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.6

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

   7. Applied rewrites 94.6%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 94.6

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

   9. Applied rewrites 94.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -5.2e-5)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (/ (pow a -1.0) (/ (- (/ c b) (/ b a)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -5.2e-5) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = pow(a, -1.0) / (((c / b) - (b / a)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -5.2e-5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(c / b) - Float64(b / a)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -5.2e-5], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.19999999999999968e-5

   1. Initial program 78.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 78.3

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

   4. Applied rewrites 78.3%

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

   if -5.19999999999999968e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 35.9%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.6

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

   7. Applied rewrites 94.6%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 94.6

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

   9. Applied rewrites 94.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - t\_0} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b \cdot b} + \frac{\frac{-1}{a}}{c}\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.16)
     (/ (pow a -1.0) (* (/ -2.0 (- (* b b) t_0)) (+ (sqrt t_0) b)))
     (/
      (pow a -1.0)
      (* (+ (/ (fma (/ a b) (/ c b) 1.0) (* b b)) (/ (/ -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 <= 1.16) {
		tmp = pow(a, -1.0) / ((-2.0 / ((b * b) - t_0)) * (sqrt(t_0) + b));
	} else {
		tmp = pow(a, -1.0) / (((fma((a / b), (c / b), 1.0) / (b * b)) + ((-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 <= 1.16)
		tmp = Float64((a ^ -1.0) / Float64(Float64(-2.0 / Float64(Float64(b * b) - t_0)) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(fma(Float64(a / b), Float64(c / b), 1.0) / Float64(b * b)) + Float64(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, 1.16], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(-2.0 / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15999999999999992

   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 85.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 85.2

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

   6. Applied rewrites 85.2%

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

   8. Applied rewrites 86.6%

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

   if 1.15999999999999992 < b

   1. Initial program 50.0%

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

   4. Applied rewrites 50.0%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 91.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b \cdot b} + \frac{\frac{-1}{a}}{c}\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - t\_0} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right) \cdot a}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.16)
     (/ (pow a -1.0) (* (/ -2.0 (- (* b b) t_0)) (+ (sqrt t_0) b)))
     (*
      (fma (/ (* (fma (* -2.0 a) c (* (- b) b)) a) (pow b 5.0)) c (/ -1.0 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 1.16) {
		tmp = pow(a, -1.0) / ((-2.0 / ((b * b) - t_0)) * (sqrt(t_0) + b));
	} else {
		tmp = fma(((fma((-2.0 * a), c, (-b * b)) * a) / pow(b, 5.0)), c, (-1.0 / b)) * 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 <= 1.16)
		tmp = Float64((a ^ -1.0) / Float64(Float64(-2.0 / Float64(Float64(b * b) - t_0)) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(fma(Float64(Float64(fma(Float64(-2.0 * a), c, Float64(Float64(-b) * b)) * a) / (b ^ 5.0)), c, Float64(-1.0 / b)) * 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, 1.16], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(-2.0 / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * c + N[((-b) * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15999999999999992

   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 85.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 85.2

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

   6. Applied rewrites 85.2%

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

   8. Applied rewrites 86.6%

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

   if 1.15999999999999992 < b

   1. Initial program 50.0%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 91.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 91.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right) \cdot a}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -5.2e-5)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (/ (fma (/ c b) (/ (* c a) b) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -5.2e-5) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = fma((c / b), ((c * a) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -5.2e-5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(c * a) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -5.2e-5], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.19999999999999968e-5

   1. Initial program 78.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 78.3

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

   4. Applied rewrites 78.3%

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

   if -5.19999999999999968e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 35.9%

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

   3. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-out N/A

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

      11. associate-/l* N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -5.2e-5)
   (/ (- (sqrt (fma (* c a) -4.0 (* b b))) b) (* 2.0 a))
   (/ (fma (/ c b) (/ (* c a) b) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -5.2e-5) {
		tmp = (sqrt(fma((c * a), -4.0, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = fma((c / b), ((c * a) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -5.2e-5)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(c * a) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -5.2e-5], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5.19999999999999968e-5

   1. Initial program 78.1%

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

   4. Applied rewrites 78.1%

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

   6. Applied rewrites 78.1%

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

   if -5.19999999999999968e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 35.9%

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

   3. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-out N/A

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

      11. associate-/l* N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -8e-6)
   (/ (- (sqrt (fma (* c a) -4.0 (* b b))) b) (* 2.0 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8e-6) {
		tmp = (sqrt(fma((c * a), -4.0, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -8e-6)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -8e-6], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -7.99999999999999964e-6

   1. Initial program 76.3%

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

   4. Applied rewrites 76.2%

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

   6. Applied rewrites 76.3%

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

   if -7.99999999999999964e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

Alternative 15: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -8e-6)
   (* (/ 0.5 a) (- (sqrt (fma (* c a) -4.0 (* b b))) b))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8e-6) {
		tmp = (0.5 / a) * (sqrt(fma((c * a), -4.0, (b * b))) - b);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -8e-6)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) - b));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -8e-6], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -7.99999999999999964e-6

   1. Initial program 76.3%

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

   4. Applied rewrites 76.2%

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

   6. Applied rewrites 76.3%

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

   if -7.99999999999999964e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

Alternative 16: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -8e-6)
   (* (/ 0.5 a) (- (sqrt (fma (* -4.0 c) a (* b b))) b))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8e-6) {
		tmp = (0.5 / a) * (sqrt(fma((-4.0 * c), a, (b * b))) - b);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -8e-6)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -8e-6], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -7.99999999999999964e-6

   1. Initial program 76.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 76.3

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

      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 76.3

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

   4. Applied rewrites 76.2%

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

   if -7.99999999999999964e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

Alternative 17: 64.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 63.9

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

5. Applied rewrites 63.9%

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

Alternative 18: 1.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 63.9

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

5. Applied rewrites 63.9%

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

7. Applied rewrites 63.8%

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

9. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2024323 
(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)))