Quadratic roots, narrow range

Percentage Accurate: 55.3% → 92.3%

Time: 15.3s

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: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ t_2 := b \cdot \left(b \cdot t\_1\right)\\ t_3 := \mathsf{fma}\left(\frac{20}{a \cdot t\_2}, -0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(t\_0 \cdot t\_0\right)\right), \frac{t\_0}{b \cdot \left(-b\right)}\right)\\ t_4 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := b + t\_5\\ t_7 := c \cdot \left(a \cdot t\_0\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) - t\_5, b \cdot b, t\_4 \cdot t\_6\right)}{t\_6 \cdot t\_6}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(t\_3 - c\right)}^{2} - \frac{\left(-2 \cdot t\_7\right) \cdot \left(2 \cdot t\_7\right)}{\left(b \cdot b\right) \cdot t\_2}}{t\_3 - \mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot -2\right)\right), c \cdot \frac{c}{t\_1}, c\right)}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c c)))
        (t_1 (* b (* b (* b b))))
        (t_2 (* b (* b t_1)))
        (t_3
         (fma
          (/ 20.0 (* a t_2))
          (* -0.25 (* (* a a) (* t_0 t_0)))
          (/ t_0 (* b (- b)))))
        (t_4 (fma c (* a -4.0) (* b b)))
        (t_5 (sqrt t_4))
        (t_6 (+ b t_5))
        (t_7 (* c (* a t_0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -8.0)
     (/ (/ (fma (- (- b) t_5) (* b b) (* t_4 t_6)) (* t_6 t_6)) (* a 2.0))
     (/
      (/
       (- (pow (- t_3 c) 2.0) (/ (* (* -2.0 t_7) (* 2.0 t_7)) (* (* b b) t_2)))
       (- t_3 (fma (* a (* a (* c -2.0))) (* c (/ c t_1)) c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * c);
	double t_1 = b * (b * (b * b));
	double t_2 = b * (b * t_1);
	double t_3 = fma((20.0 / (a * t_2)), (-0.25 * ((a * a) * (t_0 * t_0))), (t_0 / (b * -b)));
	double t_4 = fma(c, (a * -4.0), (b * b));
	double t_5 = sqrt(t_4);
	double t_6 = b + t_5;
	double t_7 = c * (a * t_0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -8.0) {
		tmp = (fma((-b - t_5), (b * b), (t_4 * t_6)) / (t_6 * t_6)) / (a * 2.0);
	} else {
		tmp = ((pow((t_3 - c), 2.0) - (((-2.0 * t_7) * (2.0 * t_7)) / ((b * b) * t_2))) / (t_3 - fma((a * (a * (c * -2.0))), (c * (c / t_1)), c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * c))
	t_1 = Float64(b * Float64(b * Float64(b * b)))
	t_2 = Float64(b * Float64(b * t_1))
	t_3 = fma(Float64(20.0 / Float64(a * t_2)), Float64(-0.25 * Float64(Float64(a * a) * Float64(t_0 * t_0))), Float64(t_0 / Float64(b * Float64(-b))))
	t_4 = fma(c, Float64(a * -4.0), Float64(b * b))
	t_5 = sqrt(t_4)
	t_6 = Float64(b + t_5)
	t_7 = Float64(c * Float64(a * t_0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -8.0)
		tmp = Float64(Float64(fma(Float64(Float64(-b) - t_5), Float64(b * b), Float64(t_4 * t_6)) / Float64(t_6 * t_6)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64((Float64(t_3 - c) ^ 2.0) - Float64(Float64(Float64(-2.0 * t_7) * Float64(2.0 * t_7)) / Float64(Float64(b * b) * t_2))) / Float64(t_3 - fma(Float64(a * Float64(a * Float64(c * -2.0))), Float64(c * Float64(c / t_1)), c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(20.0 / N[(a * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(N[(a * a), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[(b + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(c * N[(a * t$95$0), $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], -8.0], N[(N[(N[(N[((-b) - t$95$5), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(t$95$4 * t$95$6), $MachinePrecision]), $MachinePrecision] / N[(t$95$6 * t$95$6), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(t$95$3 - c), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(N[(-2.0 * t$95$7), $MachinePrecision] * N[(2.0 * t$95$7), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 - N[(N[(a * N[(a * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c / t$95$1), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $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)) < -8

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      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 89.1

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

   4. Applied egg-rr 89.1%

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

   6. Applied egg-rr 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   8. Applied egg-rr 91.0%

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

   if -8 < (/.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 52.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 94.5%

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

   7. Applied egg-rr 94.5%

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

   9. Applied egg-rr 94.7%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, b \cdot b, \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(\frac{20}{a \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, -0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right), \frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(-b\right)}\right) - c\right)}^{2} - \frac{\left(-2 \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)\right) \cdot \left(2 \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(\frac{20}{a \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}, -0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(\left(a \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right), \frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(-b\right)}\right) - \mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot -2\right)\right), c \cdot \frac{c}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c\right)}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\\ t_3 := b + t\_1\\ t_4 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ t_5 := \mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot -2\right)\right), c \cdot \frac{c}{t\_4}, c\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) - t\_1, b \cdot b, t\_0 \cdot t\_3\right)}{t\_3 \cdot t\_3}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.25, \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot \frac{20}{a \cdot \left(\left(b \cdot b\right) \cdot t\_4\right)}, \left(-c\right) \cdot \left(c \cdot \frac{a}{b \cdot b}\right)\right) + \left(\frac{\frac{\left(-2 \cdot t\_2\right) \cdot \left(2 \cdot t\_2\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot t\_4\right)\right)}}{t\_5} - \frac{c \cdot c}{t\_5}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (* c (* a (* a (* c c)))))
        (t_3 (+ b t_1))
        (t_4 (* b (* b (* b b))))
        (t_5 (fma (* a (* a (* c -2.0))) (* c (/ c t_4)) c)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -8.0)
     (/ (/ (fma (- (- b) t_1) (* b b) (* t_0 t_3)) (* t_3 t_3)) (* a 2.0))
     (/
      (+
       (fma
        -0.25
        (*
         (* (* (* a a) (* a a)) (* c (* c (* c c))))
         (/ 20.0 (* a (* (* b b) t_4))))
        (* (- c) (* c (/ a (* b b)))))
       (-
        (/ (/ (* (* -2.0 t_2) (* 2.0 t_2)) (* (* b b) (* b (* b t_4)))) t_5)
        (/ (* c c) t_5)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = c * (a * (a * (c * c)));
	double t_3 = b + t_1;
	double t_4 = b * (b * (b * b));
	double t_5 = fma((a * (a * (c * -2.0))), (c * (c / t_4)), c);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -8.0) {
		tmp = (fma((-b - t_1), (b * b), (t_0 * t_3)) / (t_3 * t_3)) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, ((((a * a) * (a * a)) * (c * (c * (c * c)))) * (20.0 / (a * ((b * b) * t_4)))), (-c * (c * (a / (b * b))))) + (((((-2.0 * t_2) * (2.0 * t_2)) / ((b * b) * (b * (b * t_4)))) / t_5) - ((c * c) / t_5))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = Float64(c * Float64(a * Float64(a * Float64(c * c))))
	t_3 = Float64(b + t_1)
	t_4 = Float64(b * Float64(b * Float64(b * b)))
	t_5 = fma(Float64(a * Float64(a * Float64(c * -2.0))), Float64(c * Float64(c / t_4)), c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -8.0)
		tmp = Float64(Float64(fma(Float64(Float64(-b) - t_1), Float64(b * b), Float64(t_0 * t_3)) / Float64(t_3 * t_3)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64(Float64(Float64(Float64(a * a) * Float64(a * a)) * Float64(c * Float64(c * Float64(c * c)))) * Float64(20.0 / Float64(a * Float64(Float64(b * b) * t_4)))), Float64(Float64(-c) * Float64(c * Float64(a / Float64(b * b))))) + Float64(Float64(Float64(Float64(Float64(-2.0 * t_2) * Float64(2.0 * t_2)) / Float64(Float64(b * b) * Float64(b * Float64(b * t_4)))) / t_5) - Float64(Float64(c * c) / t_5))) / 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]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(a * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(a * N[(a * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c / t$95$4), $MachinePrecision]), $MachinePrecision] + c), $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], -8.0], N[(N[(N[(N[((-b) - t$95$1), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(20.0 / N[(a * N[(N[(b * b), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-c) * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(-2.0 * t$95$2), $MachinePrecision] * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / t$95$5), $MachinePrecision]), $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)) < -8

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      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 89.1

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

   4. Applied egg-rr 89.1%

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

   6. Applied egg-rr 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   8. Applied egg-rr 91.0%

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

   if -8 < (/.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 52.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 94.5%

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

   7. Applied egg-rr 94.5%

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, b \cdot b, \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.25, \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot \frac{20}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \left(-c\right) \cdot \left(c \cdot \frac{a}{b \cdot b}\right)\right) + \left(\frac{\frac{\left(-2 \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)\right) \cdot \left(2 \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot -2\right)\right), c \cdot \frac{c}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c\right)} - \frac{c \cdot c}{\mathsf{fma}\left(a \cdot \left(a \cdot \left(c \cdot -2\right)\right), c \cdot \frac{c}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c\right)}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := b + t\_1\\ t_3 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) - t\_1, b \cdot b, t\_0 \cdot t\_2\right)}{t\_2 \cdot t\_2}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -2, \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{t\_3}, \mathsf{fma}\left(-0.25, \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot \frac{20}{a \cdot \left(\left(b \cdot b\right) \cdot t\_3\right)}, \left(-c\right) \cdot \left(c \cdot \frac{a}{b \cdot b}\right)\right)\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b)))
        (t_1 (sqrt t_0))
        (t_2 (+ b t_1))
        (t_3 (* b (* b (* b b)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -8.0)
     (/ (/ (fma (- (- b) t_1) (* b b) (* t_0 t_2)) (* t_2 t_2)) (* a 2.0))
     (/
      (-
       (fma
        (* c -2.0)
        (/ (* a (* a (* c c))) t_3)
        (fma
         -0.25
         (*
          (* (* (* a a) (* a a)) (* c (* c (* c c))))
          (/ 20.0 (* a (* (* b b) t_3))))
         (* (- 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 t_1 = sqrt(t_0);
	double t_2 = b + t_1;
	double t_3 = b * (b * (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -8.0) {
		tmp = (fma((-b - t_1), (b * b), (t_0 * t_2)) / (t_2 * t_2)) / (a * 2.0);
	} else {
		tmp = (fma((c * -2.0), ((a * (a * (c * c))) / t_3), fma(-0.25, ((((a * a) * (a * a)) * (c * (c * (c * c)))) * (20.0 / (a * ((b * b) * t_3)))), (-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))
	t_1 = sqrt(t_0)
	t_2 = Float64(b + t_1)
	t_3 = Float64(b * Float64(b * 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)) <= -8.0)
		tmp = Float64(Float64(fma(Float64(Float64(-b) - t_1), Float64(b * b), Float64(t_0 * t_2)) / Float64(t_2 * t_2)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(Float64(c * -2.0), Float64(Float64(a * Float64(a * Float64(c * c))) / t_3), fma(-0.25, Float64(Float64(Float64(Float64(a * a) * Float64(a * a)) * Float64(c * Float64(c * Float64(c * c)))) * Float64(20.0 / Float64(a * Float64(Float64(b * b) * t_3)))), Float64(Float64(-c) * Float64(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]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(b + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $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], -8.0], N[(N[(N[(N[((-b) - t$95$1), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * -2.0), $MachinePrecision] * N[(N[(a * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(-0.25 * N[(N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(20.0 / N[(a * N[(N[(b * b), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-c) * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -8

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      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 89.1

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

   4. Applied egg-rr 89.1%

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

   6. Applied egg-rr 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

   8. Applied egg-rr 91.0%

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

   if -8 < (/.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 52.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 94.5%

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

   7. Applied egg-rr 94.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}, b \cdot b, \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -2, \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \mathsf{fma}\left(-0.25, \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot \frac{20}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \left(-c\right) \cdot \left(c \cdot \frac{a}{b \cdot b}\right)\right)\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ t_1 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{t\_1 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -2, \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{t\_0}, \mathsf{fma}\left(-0.25, \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot \frac{20}{a \cdot \left(\left(b \cdot b\right) \cdot t\_0\right)}, \left(-c\right) \cdot \left(c \cdot \frac{a}{b \cdot b}\right)\right)\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b (* b b)))) (t_1 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -8.0)
     (/ (- t_1 (* b b)) (* (* a 2.0) (+ b (sqrt t_1))))
     (/
      (-
       (fma
        (* c -2.0)
        (/ (* a (* a (* c c))) t_0)
        (fma
         -0.25
         (*
          (* (* (* a a) (* a a)) (* c (* c (* c c))))
          (/ 20.0 (* a (* (* b b) t_0))))
         (* (- c) (* c (/ a (* b b))))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * (b * b));
	double t_1 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -8.0) {
		tmp = (t_1 - (b * b)) / ((a * 2.0) * (b + sqrt(t_1)));
	} else {
		tmp = (fma((c * -2.0), ((a * (a * (c * c))) / t_0), fma(-0.25, ((((a * a) * (a * a)) * (c * (c * (c * c)))) * (20.0 / (a * ((b * b) * t_0)))), (-c * (c * (a / (b * b)))))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	t_1 = 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)) <= -8.0)
		tmp = Float64(Float64(t_1 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_1))));
	else
		tmp = Float64(Float64(fma(Float64(c * -2.0), Float64(Float64(a * Float64(a * Float64(c * c))) / t_0), fma(-0.25, Float64(Float64(Float64(Float64(a * a) * Float64(a * a)) * Float64(c * Float64(c * Float64(c * c)))) * Float64(20.0 / Float64(a * Float64(Float64(b * b) * t_0)))), Float64(Float64(-c) * Float64(c * Float64(a / Float64(b * b)))))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -8.0], N[(N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * -2.0), $MachinePrecision] * N[(N[(a * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(-0.25 * N[(N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(20.0 / N[(a * N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-c) * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -8

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      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 89.1

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

   4. Applied egg-rr 89.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}}{2 \cdot a} \]
   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 egg-rr 90.3%

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

   if -8 < (/.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 52.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 94.5%

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

   7. Applied egg-rr 94.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -2, \frac{a \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \mathsf{fma}\left(-0.25, \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot \frac{20}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \left(-c\right) \cdot \left(c \cdot \frac{a}{b \cdot b}\right)\right)\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ t_1 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{t\_1 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{20}{a \cdot \left(\left(b \cdot b\right) \cdot t\_0\right)}, -0.25 \cdot \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right), \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \left(-2 \cdot \left(a \cdot c\right)\right)}{t\_0}\right) - \mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b (* b b)))) (t_1 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -8.0)
     (/ (- t_1 (* b b)) (* (* a 2.0) (+ b (sqrt t_1))))
     (/
      (-
       (fma
        (/ 20.0 (* a (* (* b b) t_0)))
        (* -0.25 (* (* (* a a) (* a a)) (* c (* c (* c c)))))
        (/ (* (* a (* c c)) (* -2.0 (* a c))) t_0))
       (fma c (* c (/ a (* b b))) c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * (b * b));
	double t_1 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -8.0) {
		tmp = (t_1 - (b * b)) / ((a * 2.0) * (b + sqrt(t_1)));
	} else {
		tmp = (fma((20.0 / (a * ((b * b) * t_0))), (-0.25 * (((a * a) * (a * a)) * (c * (c * (c * c))))), (((a * (c * c)) * (-2.0 * (a * c))) / t_0)) - fma(c, (c * (a / (b * b))), c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	t_1 = 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)) <= -8.0)
		tmp = Float64(Float64(t_1 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_1))));
	else
		tmp = Float64(Float64(fma(Float64(20.0 / Float64(a * Float64(Float64(b * b) * t_0))), Float64(-0.25 * Float64(Float64(Float64(a * a) * Float64(a * a)) * Float64(c * Float64(c * Float64(c * c))))), Float64(Float64(Float64(a * Float64(c * c)) * Float64(-2.0 * Float64(a * c))) / t_0)) - fma(c, Float64(c * Float64(a / Float64(b * b))), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -8.0], N[(N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(20.0 / N[(a * N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(c * N[(c * N[(a / N[(b * b), $MachinePrecision]), $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)) < -8

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      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 89.1

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

   4. Applied egg-rr 89.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}}{2 \cdot a} \]
   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 egg-rr 90.3%

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

   if -8 < (/.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 52.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 94.5%

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

   7. Applied egg-rr 94.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{20}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, -0.25 \cdot \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right), \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \left(-2 \cdot \left(a \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) - \mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -8:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, a \cdot \left(c \cdot \left(-c\right)\right)\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

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)) -8.0)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/
      (-
       (/
        (fma -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)) <= -8.0) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = ((fma(-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)) <= -8.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(fma(-2.0, Float64(Float64(Float64(a * a) * Float64(c * Float64(c * c))) / Float64(b * b)), Float64(a * Float64(c * Float64(-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], -8.0], 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[(-2.0 * N[(N[(N[(a * a), $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)) < -8

   1. Initial program 89.0%

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

      1. lift-*.f64 N/A

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

      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 89.1

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

   4. Applied egg-rr 89.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}}{2 \cdot a} \]
   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 egg-rr 90.3%

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

   if -8 < (/.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 52.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 94.5%

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

   7. Applied egg-rr 94.5%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 92.2

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

   10. Simplified 92.2%

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

4. Final simplification 92.0%

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

Alternative 7: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

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)) -0.009)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/ (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)) <= -0.009) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = 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)) <= -0.009)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-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], -0.009], 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[(c * c), $MachinePrecision] * N[(a / 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)) < -0.00899999999999999932

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 80.8

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

   4. Applied egg-rr 80.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}}{2 \cdot a} \]
   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 egg-rr 82.0%

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

   if -0.00899999999999999932 < (/.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 47.2%

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

   3. Taylor expanded in 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 90.4

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

   5. Simplified 90.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.009)
   (/ (- (sqrt (fma b b (* a (* c -4.0)))) b) (* a 2.0))
   (/ (fma (* c c) (/ a (* 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)) <= -0.009) {
		tmp = (sqrt(fma(b, b, (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma((c * c), (a / (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)) <= -0.009)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-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], -0.009], 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[(N[(N[(c * c), $MachinePrecision] * N[(a / 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)) < -0.00899999999999999932

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 80.8

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

   4. Applied egg-rr 80.8%

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

   if -0.00899999999999999932 < (/.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 47.2%

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

   3. Taylor expanded in 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 90.4

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

   5. Simplified 90.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.009)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
   (/ (fma (* c c) (/ a (* 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)) <= -0.009) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = fma((c * c), (a / (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)) <= -0.009)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-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], -0.009], 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[(N[(N[(c * c), $MachinePrecision] * N[(a / 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)) < -0.00899999999999999932

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 80.7

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

   4. Applied egg-rr 80.7%

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

   if -0.00899999999999999932 < (/.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 47.2%

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

   3. Taylor expanded in 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 90.4

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

   5. Simplified 90.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.009)
   (* (/ -0.5 a) (- b (sqrt (fma a (* c -4.0) (* b b)))))
   (/ (fma (* c c) (/ a (* 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)) <= -0.009) {
		tmp = (-0.5 / a) * (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else {
		tmp = fma((c * c), (a / (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)) <= -0.009)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-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], -0.009], 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[(N[(N[(c * c), $MachinePrecision] * N[(a / 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)) < -0.00899999999999999932

   1. Initial program 80.7%

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

   4. Applied egg-rr 80.7%

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

   if -0.00899999999999999932 < (/.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 47.2%

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

   3. Taylor expanded in 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 90.4

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

   5. Simplified 90.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% 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 56.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
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 82.6

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

5. Simplified 82.6%

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

6. Final simplification 82.6%

   \[\leadsto \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b} \]
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 56.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 64.4

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

5. Simplified 64.4%

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

Reproduce

herbie shell --seed 2024208 
(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)))