Quadratic roots, narrow range

Percentage Accurate: 55.0% → 92.1%

Time: 14.9s

Alternatives: 12

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: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot c\right)\\ t_1 := a \cdot t\_0\\ t_2 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.5:\\ \;\;\;\;\frac{\frac{-1}{\frac{b + \sqrt{t\_2}}{b \cdot b - t\_2}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-5 \cdot \left(\left(a \cdot c\right) \cdot \left(c \cdot \left(a \cdot t\_1\right)\right)\right)}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} + \frac{\frac{c \cdot \left(t\_1 \cdot -2\right)}{b \cdot b} - t\_0}{b \cdot b}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a c))) (t_1 (* a t_0)) (t_2 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -1.5)
     (/ (/ -1.0 (/ (+ b (sqrt t_2)) (- (* b b) t_2))) (* a 2.0))
     (/
      (-
       (+
        (/
         (* -5.0 (* (* a c) (* c (* a t_1))))
         (* a (* (* b b) (* b (* b (* b b))))))
        (/ (- (/ (* c (* t_1 -2.0)) (* b b)) t_0) (* b b)))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * c);
	double t_1 = a * t_0;
	double t_2 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.5) {
		tmp = (-1.0 / ((b + sqrt(t_2)) / ((b * b) - t_2))) / (a * 2.0);
	} else {
		tmp = ((((-5.0 * ((a * c) * (c * (a * t_1)))) / (a * ((b * b) * (b * (b * (b * b)))))) + ((((c * (t_1 * -2.0)) / (b * b)) - t_0) / (b * b))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * c))
	t_1 = Float64(a * t_0)
	t_2 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.5)
		tmp = Float64(Float64(-1.0 / Float64(Float64(b + sqrt(t_2)) / Float64(Float64(b * b) - t_2))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-5.0 * Float64(Float64(a * c) * Float64(c * Float64(a * t_1)))) / Float64(a * Float64(Float64(b * b) * Float64(b * Float64(b * Float64(b * b)))))) + Float64(Float64(Float64(Float64(c * Float64(t_1 * -2.0)) / Float64(b * b)) - t_0) / Float64(b * b))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(-1.0 / N[(N[(b + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-5.0 * N[(N[(a * c), $MachinePrecision] * N[(c * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $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)) < -1.5

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 86.2

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

   4. Applied rewrites 86.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 87.3%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{c}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(\left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right) \cdot \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right)\right)}{\left(a \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)\right)\right) \cdot 0.05} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}}{b} \]

   9. Applied rewrites 94.6%

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

   11. Applied rewrites 94.6%

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

4. Final simplification 93.5%

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

Alternative 2: 89.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.5

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 86.2

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

   4. Applied rewrites 86.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 87.3%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.2%

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

4. Final simplification 92.3%

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

Alternative 3: 89.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.5

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 86.2

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

   4. Applied rewrites 86.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-+ N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.2%

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

4. Final simplification 92.3%

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

Alternative 4: 89.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 86.2

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

   4. Applied rewrites 86.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-+ N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{c}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(\left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right) \cdot \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right)\right)}{\left(a \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)\right)\right) \cdot 0.05} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}}{b} \]

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 93.2%

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

4. Final simplification 92.3%

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

Alternative 5: 89.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.5) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = ((((((-2.0 * (a * a)) * (c * (c * c))) / (b * b)) - (a * (c * c))) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.5)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(a * a)) * Float64(c * Float64(c * c))) / Float64(b * b)) - Float64(a * Float64(c * c))) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

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

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 86.2

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

   4. Applied rewrites 86.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-+ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 87.3%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right) \cdot \frac{c}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(\left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right) \cdot \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right)\right)}{\left(a \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)\right)\right) \cdot 0.05} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}}{b} \]

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 93.2%

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

4. Final simplification 92.3%

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

Alternative 6: 85.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.5) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.5)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

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

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 86.2

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

   4. Applied rewrites 86.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-+ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 87.3%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.6%

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

   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

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 88.9

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

   8. Applied rewrites 88.9%

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

4. Final simplification 88.6%

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

Alternative 7: 84.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.5) {
		tmp = (sqrt(fma(b, b, (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $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)) < -1.5

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 86.2

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

   4. Applied rewrites 86.2%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.6%

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

   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

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 88.9

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

   8. Applied rewrites 88.9%

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

4. Final simplification 88.5%

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

Alternative 8: 84.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.5) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.5)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $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)) < -1.5

   1. Initial program 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.1

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

   4. Applied rewrites 86.1%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.6%

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

   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

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 88.9

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

   8. Applied rewrites 88.9%

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

4. Final simplification 88.5%

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

Alternative 9: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -1.5:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -1.5)
   (* (/ -0.5 a) (- b (sqrt (fma a (* c -4.0) (* b b)))))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -1.5) {
		tmp = (-0.5 / a) * (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -1.5)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $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)) < -1.5

   1. Initial program 86.1%

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

   4. Applied rewrites 86.1%

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

   if -1.5 < (/.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 44.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.6%

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

   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

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 88.9

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

   8. Applied rewrites 88.9%

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

4. Final simplification 88.5%

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

Alternative 10: 81.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 91.6%

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

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

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 83.6

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

8. Applied rewrites 83.6%

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

Alternative 11: 81.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 83.6

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

5. Applied rewrites 83.6%

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

6. Final simplification 83.6%

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

Alternative 12: 64.6% 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(c / Float64(-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 50.3%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 68.3

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

5. Applied rewrites 68.3%

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

Reproduce

herbie shell --seed 2024216 
(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)))