quadp (p42, positive)

Percentage Accurate: 52.0% → 70.2%

Time: 12.4s

Alternatives: 11

Speedup: 0.9×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.0% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 70.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a}, t\_0, \frac{b}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4, a \cdot c, 0\right)}{a \cdot 2}}{b + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* a c) (* b b)))))
   (if (<= b -1.35e+154)
     (* -0.5 (/ b a))
     (if (<= b 2e-299)
       (fma (/ 0.5 a) t_0 (/ b (* a -2.0)))
       (/ (/ (fma -4.0 (* a c) 0.0) (* a 2.0)) (+ b t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (a * c), (b * b)));
	double tmp;
	if (b <= -1.35e+154) {
		tmp = -0.5 * (b / a);
	} else if (b <= 2e-299) {
		tmp = fma((0.5 / a), t_0, (b / (a * -2.0)));
	} else {
		tmp = (fma(-4.0, (a * c), 0.0) / (a * 2.0)) / (b + t_0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(a * c), Float64(b * b)))
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 2e-299)
		tmp = fma(Float64(0.5 / a), t_0, Float64(b / Float64(a * -2.0)));
	else
		tmp = Float64(Float64(fma(-4.0, Float64(a * c), 0.0) / Float64(a * 2.0)) / Float64(b + t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.35e+154], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-299], N[(N[(0.5 / a), $MachinePrecision] * t$95$0 + N[(b / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.0 * N[(a * c), $MachinePrecision] + 0.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 29.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.0

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 1.99999999999999998e-299

   1. Initial program 90.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 90.5

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

   4. Applied egg-rr 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 90.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 90.3

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 90.3

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

   6. Applied egg-rr 90.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. div-inv N/A

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

      13. distribute-neg-frac N/A

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

      14. lift-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

   8. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{a}, \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}, \frac{b}{a \cdot -2}\right)} \]

   if 1.99999999999999998e-299 < b

   1. Initial program 26.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 26.5

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

   4. Applied egg-rr 26.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 26.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 26.5

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 26.5

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

   6. Applied egg-rr 26.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 26.5

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

   8. Applied egg-rr 26.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip-- N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/l/ N/A

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

   10. Applied egg-rr 66.7%

       \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4, a \cdot c, 0\right)}{a \cdot 2}}{b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+154)
   (* -0.5 (/ b a))
   (if (<= b 3.2e-101)
     (- (/ (sqrt (fma a (* -4.0 c) (* b b))) (* a 2.0)) (/ b (* a 2.0)))
     (/
      (* -4.0 (* a c))
      (* (* a 2.0) (+ b (sqrt (fma -4.0 (* a c) (* b b)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+154) {
		tmp = -0.5 * (b / a);
	} else if (b <= 3.2e-101) {
		tmp = (sqrt(fma(a, (-4.0 * c), (b * b))) / (a * 2.0)) - (b / (a * 2.0));
	} else {
		tmp = (-4.0 * (a * c)) / ((a * 2.0) * (b + sqrt(fma(-4.0, (a * c), (b * b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 3.2e-101)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * b))) / Float64(a * 2.0)) - Float64(b / Float64(a * 2.0)));
	else
		tmp = Float64(Float64(-4.0 * Float64(a * c)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(fma(-4.0, Float64(a * c), Float64(b * b))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+154], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-101], N[(N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] - N[(b / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 29.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.0

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 3.19999999999999978e-101

   1. Initial program 86.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lift-*.f64 N/A

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

      11. div-sub N/A

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

      12. lower--.f64 N/A

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

   4. Applied egg-rr 86.7%

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

   if 3.19999999999999978e-101 < b

   1. Initial program 13.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 13.9

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

   4. Applied egg-rr 13.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 12.3%

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

   7. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 59.8

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

   9. Simplified 59.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a}, t\_0, \frac{b}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(b + t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* a c) (* b b)))))
   (if (<= b -1.35e+154)
     (* -0.5 (/ b a))
     (if (<= b 3.2e-101)
       (fma (/ 0.5 a) t_0 (/ b (* a -2.0)))
       (/ (* -4.0 (* a c)) (* (* a 2.0) (+ b t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (a * c), (b * b)));
	double tmp;
	if (b <= -1.35e+154) {
		tmp = -0.5 * (b / a);
	} else if (b <= 3.2e-101) {
		tmp = fma((0.5 / a), t_0, (b / (a * -2.0)));
	} else {
		tmp = (-4.0 * (a * c)) / ((a * 2.0) * (b + t_0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(a * c), Float64(b * b)))
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 3.2e-101)
		tmp = fma(Float64(0.5 / a), t_0, Float64(b / Float64(a * -2.0)));
	else
		tmp = Float64(Float64(-4.0 * Float64(a * c)) / Float64(Float64(a * 2.0) * Float64(b + t_0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.35e+154], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-101], N[(N[(0.5 / a), $MachinePrecision] * t$95$0 + N[(b / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 29.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.0

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 3.19999999999999978e-101

   1. Initial program 86.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 86.6

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

   4. Applied egg-rr 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 86.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 86.5

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 86.5

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

   6. Applied egg-rr 86.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. div-inv N/A

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

      13. distribute-neg-frac N/A

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

      14. lift-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

   8. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{a}, \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}, \frac{b}{a \cdot -2}\right)} \]

   if 3.19999999999999978e-101 < b

   1. Initial program 13.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 13.9

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

   4. Applied egg-rr 13.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 12.3%

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

   7. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 59.8

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

   9. Simplified 59.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a}, \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}, \frac{b}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot 4\right) \cdot \frac{1}{\left(a \cdot 2\right) \cdot \left(b - t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma a (* -4.0 c) (* b b)))))
   (if (<= b -1.35e+154)
     (* -0.5 (/ b a))
     (if (<= b 4.5e+155)
       (/ (- t_0 b) (* a 2.0))
       (* (* c 4.0) (/ 1.0 (* (* a 2.0) (- b t_0))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(a, (-4.0 * c), (b * b)));
	double tmp;
	if (b <= -1.35e+154) {
		tmp = -0.5 * (b / a);
	} else if (b <= 4.5e+155) {
		tmp = (t_0 - b) / (a * 2.0);
	} else {
		tmp = (c * 4.0) * (1.0 / ((a * 2.0) * (b - t_0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(a, Float64(-4.0 * c), Float64(b * b)))
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 4.5e+155)
		tmp = Float64(Float64(t_0 - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c * 4.0) * Float64(1.0 / Float64(Float64(a * 2.0) * Float64(b - t_0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.35e+154], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e+155], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 4.0), $MachinePrecision] * N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 29.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.0

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 4.49999999999999973e155

   1. Initial program 65.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 65.4

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

   4. Applied egg-rr 65.4%

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

   if 4.49999999999999973e155 < b

   1. Initial program 1.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 33.9

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

   7. Simplified 33.9%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot 4\right) \cdot \frac{1}{\left(a \cdot 2\right) \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{+127}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot -4}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+154)
   (* -0.5 (/ b a))
   (if (<= b 3.7e+127)
     (/ (- (sqrt (fma a (* -4.0 c) (* b b))) b) (* a 2.0))
     (/ (* a -4.0) (* (* a 2.0) (+ b (sqrt (fma -4.0 (* a c) (* b b)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+154) {
		tmp = -0.5 * (b / a);
	} else if (b <= 3.7e+127) {
		tmp = (sqrt(fma(a, (-4.0 * c), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = (a * -4.0) / ((a * 2.0) * (b + sqrt(fma(-4.0, (a * c), (b * b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 3.7e+127)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * -4.0) / Float64(Float64(a * 2.0) * Float64(b + sqrt(fma(-4.0, Float64(a * c), Float64(b * b))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+154], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.7e+127], N[(N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * -4.0), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 29.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.0

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 3.6999999999999998e127

   1. Initial program 69.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 69.7

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

   4. Applied egg-rr 69.7%

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

   if 3.6999999999999998e127 < b

   1. Initial program 5.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 5.2

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

   4. Applied egg-rr 5.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied egg-rr 4.1%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 30.1

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

   9. Simplified 30.1%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{+127}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot -4}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+154)
   (* -0.5 (/ b a))
   (if (<= b 1.6e+143)
     (/ (- (sqrt (fma a (* -4.0 c) (* b b))) b) (* a 2.0))
     (/ (* (* a c) -2.0) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+154) {
		tmp = -0.5 * (b / a);
	} else if (b <= 1.6e+143) {
		tmp = (sqrt(fma(a, (-4.0 * c), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = ((a * c) * -2.0) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 1.6e+143)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * c) * -2.0) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+154], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+143], N[(N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 29.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.0

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 1.60000000000000008e143

   1. Initial program 67.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 67.6

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

   4. Applied egg-rr 67.6%

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

   if 1.60000000000000008e143 < b

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 15.2

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

   7. Simplified 15.2%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+154)
   (* -0.5 (/ b a))
   (if (<= b 1.6e+143)
     (* (/ 0.5 a) (- (sqrt (fma -4.0 (* a c) (* b b))) b))
     (/ (* (* a c) -2.0) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+154) {
		tmp = -0.5 * (b / a);
	} else if (b <= 1.6e+143) {
		tmp = (0.5 / a) * (sqrt(fma(-4.0, (a * c), (b * b))) - b);
	} else {
		tmp = ((a * c) * -2.0) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 1.6e+143)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(-4.0, Float64(a * c), Float64(b * b))) - b));
	else
		tmp = Float64(Float64(Float64(a * c) * -2.0) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+154], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+143], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 29.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.0

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 1.60000000000000008e143

   1. Initial program 67.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 67.6

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

   4. Applied egg-rr 67.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 67.5

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

   6. Applied egg-rr 67.5%

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

   if 1.60000000000000008e143 < b

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 15.2

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

   7. Simplified 15.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+154)
   (* -0.5 (/ b a))
   (if (<= b 1.6e+143)
     (* (- b (sqrt (fma a (* -4.0 c) (* b b)))) (/ -0.5 a))
     (/ (* (* a c) -2.0) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e+154) {
		tmp = -0.5 * (b / a);
	} else if (b <= 1.6e+143) {
		tmp = (b - sqrt(fma(a, (-4.0 * c), (b * b)))) * (-0.5 / a);
	} else {
		tmp = ((a * c) * -2.0) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e+154)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 1.6e+143)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(-4.0 * c), Float64(b * b)))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(Float64(a * c) * -2.0) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+154], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+143], N[(N[(b - N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 29.4%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.0

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.35000000000000003e154 < b < 1.60000000000000008e143

   1. Initial program 67.6%

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

   4. Applied egg-rr 67.5%

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

   if 1.60000000000000008e143 < b

   1. Initial program 1.7%

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 15.2

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

   7. Simplified 15.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+143}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+53}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e+53)
   (* -0.5 (/ b a))
   (if (<= b 1.04e+45)
     (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0))
     (/ (* (* a c) -2.0) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e+53) {
		tmp = -0.5 * (b / a);
	} else if (b <= 1.04e+45) {
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = ((a * c) * -2.0) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.3d+53)) then
        tmp = (-0.5d0) * (b / a)
    else if (b <= 1.04d+45) then
        tmp = (sqrt(((-4.0d0) * (a * c))) - b) / (a * 2.0d0)
    else
        tmp = ((a * c) * (-2.0d0)) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e+53) {
		tmp = -0.5 * (b / a);
	} else if (b <= 1.04e+45) {
		tmp = (Math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = ((a * c) * -2.0) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.3e+53:
		tmp = -0.5 * (b / a)
	elif b <= 1.04e+45:
		tmp = (math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0)
	else:
		tmp = ((a * c) * -2.0) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.3e+53)
		tmp = Float64(-0.5 * Float64(b / a));
	elseif (b <= 1.04e+45)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * c) * -2.0) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.3e+53)
		tmp = -0.5 * (b / a);
	elseif (b <= 1.04e+45)
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	else
		tmp = ((a * c) * -2.0) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.3e+53], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.04e+45], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.29999999999999999e53

   1. Initial program 53.6%

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

   4. Applied egg-rr 0.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 40.7

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 40.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -1.29999999999999999e53 < b < 1.04e45

   1. Initial program 72.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 72.8

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

   4. Applied egg-rr 72.8%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.4

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

   7. Simplified 56.4%

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

   if 1.04e45 < b

   1. Initial program 7.5%

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

   4. Applied egg-rr 0.3%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 14.6

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

   7. Simplified 14.6%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+53}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 19.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-164}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-164) (* -0.5 (/ b a)) (/ (* (* a c) -2.0) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-164) {
		tmp = -0.5 * (b / a);
	} else {
		tmp = ((a * c) * -2.0) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-164)) then
        tmp = (-0.5d0) * (b / a)
    else
        tmp = ((a * c) * (-2.0d0)) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-164) {
		tmp = -0.5 * (b / a);
	} else {
		tmp = ((a * c) * -2.0) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-164:
		tmp = -0.5 * (b / a)
	else:
		tmp = ((a * c) * -2.0) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-164)
		tmp = Float64(-0.5 * Float64(b / a));
	else
		tmp = Float64(Float64(Float64(a * c) * -2.0) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-164)
		tmp = -0.5 * (b / a);
	else
		tmp = ((a * c) * -2.0) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-164], N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.99999999999999962e-164

   1. Initial program 68.6%

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

   4. Applied egg-rr 20.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 30.4

         \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

   7. Simplified 30.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]

   if -4.99999999999999962e-164 < b

   1. Initial program 33.9%

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

   4. Applied egg-rr 12.7%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 11.4

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

   7. Simplified 11.4%

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

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-164}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 15.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.5 * (b / 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 = (-0.5d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (b / a);
}

Python

def code(a, b, c):
	return -0.5 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.0%

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

4. Applied egg-rr 15.7%

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

5. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 14.1

      \[\leadsto -0.5 \cdot \color{blue}{\frac{b}{a}} \]

7. Simplified 14.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024214 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))