quadp (p42, positive)

Percentage Accurate: 52.1% → 90.5%

Time: 11.5s

Alternatives: 13

Speedup: 2.5×

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: 52.1% 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: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+106}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{a}, 0.5, \frac{b}{a \cdot -2}\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{c \cdot -2}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma a (* c -4.0) (* b b)))))
   (if (<= b -2e+106)
     (- (/ b a))
     (if (<= b 4.2e-296)
       (fma (/ t_0 a) 0.5 (/ b (* a -2.0)))
       (if (<= b 1.6e+84)
         (/ (* c -2.0) (+ b t_0))
         (/ (* c -2.0) (fma c (/ (* a -2.0) b) (* b 2.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(a, (c * -4.0), (b * b)));
	double tmp;
	if (b <= -2e+106) {
		tmp = -(b / a);
	} else if (b <= 4.2e-296) {
		tmp = fma((t_0 / a), 0.5, (b / (a * -2.0)));
	} else if (b <= 1.6e+84) {
		tmp = (c * -2.0) / (b + t_0);
	} else {
		tmp = (c * -2.0) / fma(c, ((a * -2.0) / b), (b * 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))
	tmp = 0.0
	if (b <= -2e+106)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 4.2e-296)
		tmp = fma(Float64(t_0 / a), 0.5, Float64(b / Float64(a * -2.0)));
	elseif (b <= 1.6e+84)
		tmp = Float64(Float64(c * -2.0) / Float64(b + t_0));
	else
		tmp = Float64(Float64(c * -2.0) / fma(c, Float64(Float64(a * -2.0) / b), Float64(b * 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2e+106], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 4.2e-296], N[(N[(t$95$0 / a), $MachinePrecision] * 0.5 + N[(b / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+84], N[(N[(c * -2.0), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.00000000000000018e106

   1. Initial program 50.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.00000000000000018e106 < b < 4.1999999999999999e-296

   1. Initial program 81.0%

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

   4. Applied rewrites 81.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

      11. lift--.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. lift-*.f64 N/A

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

      17. div-sub N/A

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

   6. Applied rewrites 81.2%

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

   if 4.1999999999999999e-296 < b < 1.60000000000000005e84

   1. Initial program 53.5%

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

   4. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 80.5%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 89.3

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

   9. Applied rewrites 89.3%

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

   if 1.60000000000000005e84 < b

   1. Initial program 4.2%

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

   4. Applied rewrites 4.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 57.7

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 96.6

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

   12. Applied rewrites 96.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+106}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}, 0.5, \frac{b}{a \cdot -2}\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+106}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq -7.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+106)
   (- (/ b a))
   (if (<= b -7.8e-124)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b 1.6e+84)
       (/ (* c -2.0) (+ b (sqrt (fma a (* c -4.0) (* b b)))))
       (/ (* c -2.0) (fma c (/ (* a -2.0) b) (* b 2.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+106) {
		tmp = -(b / a);
	} else if (b <= -7.8e-124) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 1.6e+84) {
		tmp = (c * -2.0) / (b + sqrt(fma(a, (c * -4.0), (b * b))));
	} else {
		tmp = (c * -2.0) / fma(c, ((a * -2.0) / b), (b * 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+106)
		tmp = Float64(-Float64(b / a));
	elseif (b <= -7.8e-124)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= 1.6e+84)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	else
		tmp = Float64(Float64(c * -2.0) / fma(c, Float64(Float64(a * -2.0) / b), Float64(b * 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+106], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, -7.8e-124], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+84], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.00000000000000018e106

   1. Initial program 50.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.00000000000000018e106 < b < -7.79999999999999986e-124

   1. Initial program 91.6%

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

   if -7.79999999999999986e-124 < b < 1.60000000000000005e84

   1. Initial program 57.1%

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

   4. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 75.6%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 82.1

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

   9. Applied rewrites 82.1%

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

   if 1.60000000000000005e84 < b

   1. Initial program 4.2%

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

   4. Applied rewrites 4.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 57.7

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 96.6

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

   12. Applied rewrites 96.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+106}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq -7.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+106}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - t\_0\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{c \cdot -2}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma a (* c -4.0) (* b b)))))
   (if (<= b -2.2e+106)
     (- (/ b a))
     (if (<= b -6.5e-274)
       (* (/ -0.5 a) (- b t_0))
       (if (<= b 1.6e+84)
         (/ (* c -2.0) (+ b t_0))
         (/ (* c -2.0) (fma c (/ (* a -2.0) b) (* b 2.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(a, (c * -4.0), (b * b)));
	double tmp;
	if (b <= -2.2e+106) {
		tmp = -(b / a);
	} else if (b <= -6.5e-274) {
		tmp = (-0.5 / a) * (b - t_0);
	} else if (b <= 1.6e+84) {
		tmp = (c * -2.0) / (b + t_0);
	} else {
		tmp = (c * -2.0) / fma(c, ((a * -2.0) / b), (b * 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))
	tmp = 0.0
	if (b <= -2.2e+106)
		tmp = Float64(-Float64(b / a));
	elseif (b <= -6.5e-274)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - t_0));
	elseif (b <= 1.6e+84)
		tmp = Float64(Float64(c * -2.0) / Float64(b + t_0));
	else
		tmp = Float64(Float64(c * -2.0) / fma(c, Float64(Float64(a * -2.0) / b), Float64(b * 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.2e+106], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, -6.5e-274], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+84], N[(N[(c * -2.0), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.19999999999999992e106

   1. Initial program 50.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.19999999999999992e106 < b < -6.49999999999999959e-274

   1. Initial program 81.6%

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

   4. Applied rewrites 81.7%

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

   if -6.49999999999999959e-274 < b < 1.60000000000000005e84

   1. Initial program 55.7%

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

   4. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 79.9%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 88.0

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

   9. Applied rewrites 88.0%

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

   if 1.60000000000000005e84 < b

   1. Initial program 4.2%

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

   4. Applied rewrites 4.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 57.0%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 57.7

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 96.6

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

   12. Applied rewrites 96.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+106}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+106}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+106)
   (- (/ b a))
   (if (<= b 2.5e-89)
     (* (/ -0.5 a) (- b (sqrt (fma a (* c -4.0) (* b b)))))
     (/ (* c -2.0) (fma c (/ (* a -2.0) b) (* b 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+106) {
		tmp = -(b / a);
	} else if (b <= 2.5e-89) {
		tmp = (-0.5 / a) * (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else {
		tmp = (c * -2.0) / fma(c, ((a * -2.0) / b), (b * 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+106)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 2.5e-89)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	else
		tmp = Float64(Float64(c * -2.0) / fma(c, Float64(Float64(a * -2.0) / b), Float64(b * 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+106], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 2.5e-89], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.19999999999999992e106

   1. Initial program 50.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.19999999999999992e106 < b < 2.49999999999999983e-89

   1. Initial program 80.4%

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

   4. Applied rewrites 80.5%

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

   if 2.49999999999999983e-89 < b

   1. Initial program 18.9%

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

   4. Applied rewrites 18.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 68.1%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 73.6

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

   9. Applied rewrites 73.6%

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

   10. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 84.7

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

   12. Applied rewrites 84.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+106}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-86)
   (fma b (/ c (* b b)) (- (/ b a)))
   (if (<= b 2.5e-89)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (* c -2.0) (fma c (/ (* a -2.0) b) (* b 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-86) {
		tmp = fma(b, (c / (b * b)), -(b / a));
	} else if (b <= 2.5e-89) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (c * -2.0) / fma(c, ((a * -2.0) / b), (b * 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-86)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	elseif (b <= 2.5e-89)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c * -2.0) / fma(c, Float64(Float64(a * -2.0) / b), Float64(b * 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-86], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 2.5e-89], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e-86

   1. Initial program 68.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.0000000000000001e-86 < b < 2.49999999999999983e-89

   1. Initial program 73.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if 2.49999999999999983e-89 < b

   1. Initial program 18.9%

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

   4. Applied rewrites 18.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 68.1%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 73.6

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

   9. Applied rewrites 73.6%

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

   10. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 84.7

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

   12. Applied rewrites 84.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-86)
   (fma b (/ c (* b b)) (- (/ b a)))
   (if (<= b 2.5e-89)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (* c -2.0) (+ b (fma c (/ (* a -2.0) b) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-86) {
		tmp = fma(b, (c / (b * b)), -(b / a));
	} else if (b <= 2.5e-89) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (c * -2.0) / (b + fma(c, ((a * -2.0) / b), b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-86)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	elseif (b <= 2.5e-89)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c * -2.0) / Float64(b + fma(c, Float64(Float64(a * -2.0) / b), b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-86], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 2.5e-89], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e-86

   1. Initial program 68.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.0000000000000001e-86 < b < 2.49999999999999983e-89

   1. Initial program 73.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if 2.49999999999999983e-89 < b

   1. Initial program 18.9%

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

   4. Applied rewrites 18.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 68.1%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 73.6

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

   9. Applied rewrites 73.6%

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

   10. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r/ N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 84.7

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

   12. Applied rewrites 84.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-86)
   (fma b (/ c (* b b)) (- (/ b a)))
   (if (<= b 2.5e-89)
     (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (/ (* c -2.0) (+ b (fma c (/ (* a -2.0) b) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-86) {
		tmp = fma(b, (c / (b * b)), -(b / a));
	} else if (b <= 2.5e-89) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} else {
		tmp = (c * -2.0) / (b + fma(c, ((a * -2.0) / b), b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-86)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	elseif (b <= 2.5e-89)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(c * -2.0) / Float64(b + fma(c, Float64(Float64(a * -2.0) / b), b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-86], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 2.5e-89], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e-86

   1. Initial program 68.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.0000000000000001e-86 < b < 2.49999999999999983e-89

   1. Initial program 73.9%

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

   4. Applied rewrites 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

   6. Applied rewrites 71.9%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \color{blue}{\frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}\right) \]

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0

         \[\leadsto \frac{-0.5}{a} \cdot \left(b - \frac{1}{\sqrt{\frac{-0.25}{\color{blue}{c \cdot a}}}}\right) \]

   9. Applied rewrites 69.0%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \frac{1}{\sqrt{\color{blue}{\frac{-0.25}{c \cdot a}}}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift--.f64 69.0

          \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(b - \frac{1}{\sqrt{\frac{-0.25}{c \cdot a}}}\right)} \]

       6. lift-/.f64 N/A

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

       7. metadata-eval N/A

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

       8. lift-sqrt.f64 N/A

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

       9. sqrt-div N/A

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

       10. lift-/.f64 N/A

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

       11. clear-num N/A

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

       12. div-inv N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. metadata-eval N/A

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

       16. associate-*r* N/A

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

       17. lift-*.f64 N/A

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

       18. lower-sqrt.f64 N/A

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

       19. lower-*.f64 71.8

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

   11. Applied rewrites 71.8%

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

   if 2.49999999999999983e-89 < b

   1. Initial program 18.9%

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

   4. Applied rewrites 18.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift--.f64 N/A

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

      9. flip-- N/A

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

      10. lift-+.f64 N/A

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

      11. associate-*l/ N/A

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

   6. Applied rewrites 68.1%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 73.6

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

   9. Applied rewrites 73.6%

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

   10. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r/ N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 84.7

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

   12. Applied rewrites 84.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(c, \frac{a \cdot -2}{b}, b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-86)
   (fma b (/ c (* b b)) (- (/ b a)))
   (if (<= b 2.5e-89)
     (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-86) {
		tmp = fma(b, (c / (b * b)), -(b / a));
	} else if (b <= 2.5e-89) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-86)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	elseif (b <= 2.5e-89)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-86], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 2.5e-89], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e-86

   1. Initial program 68.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.0000000000000001e-86 < b < 2.49999999999999983e-89

   1. Initial program 73.9%

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

   4. Applied rewrites 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

   6. Applied rewrites 71.9%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \color{blue}{\frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}\right) \]

   7. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0

         \[\leadsto \frac{-0.5}{a} \cdot \left(b - \frac{1}{\sqrt{\frac{-0.25}{\color{blue}{c \cdot a}}}}\right) \]

   9. Applied rewrites 69.0%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \frac{1}{\sqrt{\color{blue}{\frac{-0.25}{c \cdot a}}}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift--.f64 69.0

          \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(b - \frac{1}{\sqrt{\frac{-0.25}{c \cdot a}}}\right)} \]

       6. lift-/.f64 N/A

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

       7. metadata-eval N/A

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

       8. lift-sqrt.f64 N/A

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

       9. sqrt-div N/A

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

       10. lift-/.f64 N/A

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

       11. clear-num N/A

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

       12. div-inv N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. metadata-eval N/A

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

       16. associate-*r* N/A

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

       17. lift-*.f64 N/A

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

       18. lower-sqrt.f64 N/A

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

       19. lower-*.f64 71.8

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

   11. Applied rewrites 71.8%

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

   if 2.49999999999999983e-89 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-86)
   (fma b (/ c (* b b)) (- (/ b a)))
   (if (<= b 2.5e-89)
     (* (/ -0.5 a) (- b (sqrt (* -4.0 (* a c)))))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-86) {
		tmp = fma(b, (c / (b * b)), -(b / a));
	} else if (b <= 2.5e-89) {
		tmp = (-0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-86)
		tmp = fma(b, Float64(c / Float64(b * b)), Float64(-Float64(b / a)));
	elseif (b <= 2.5e-89)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(-4.0 * Float64(a * c)))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-86], N[(b * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-N[(b / a), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 2.5e-89], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e-86

   1. Initial program 68.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -3.0000000000000001e-86 < b < 2.49999999999999983e-89

   1. Initial program 73.9%

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 71.6

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

   7. Applied rewrites 71.6%

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

   if 2.49999999999999983e-89 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 84.7

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

   5. Applied rewrites 84.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{c}{b \cdot b}, -\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-286}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.25e-286) (- (/ b a)) (- (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.25e-286) {
		tmp = -(b / a);
	} else {
		tmp = -(c / b);
	}
	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 <= 1.25d-286) then
        tmp = -(b / a)
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.25e-286) {
		tmp = -(b / a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.25e-286:
		tmp = -(b / a)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.25e-286)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.25e-286)
		tmp = -(b / a);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.25e-286], (-N[(b / a), $MachinePrecision]), (-N[(c / b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 1.25000000000000009e-286

   1. Initial program 69.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if 1.25000000000000009e-286 < b

   1. Initial program 32.0%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 69.2

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

   5. Applied rewrites 69.2%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-286}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5200000000000:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5200000000000.0) (- (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5200000000000.0) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	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 <= 5200000000000.0d0) then
        tmp = -(b / a)
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5200000000000.0) {
		tmp = -(b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5200000000000.0:
		tmp = -(b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5200000000000.0)
		tmp = Float64(-Float64(b / a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5200000000000.0)
		tmp = -(b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5200000000000.0], (-N[(b / a), $MachinePrecision]), N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2e12

   1. Initial program 68.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 43.0

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

   5. Applied rewrites 43.0%

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

   if 5.2e12 < b

   1. Initial program 12.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 92.8

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

   5. Applied rewrites 92.8%

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

      1. neg-sub0 N/A

         \[\leadsto \frac{c}{\color{blue}{0 - b}} \]

      2. flip3-- N/A

         \[\leadsto \frac{c}{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{c}{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      4. div-sub N/A

         \[\leadsto \frac{c}{\color{blue}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{b}^{\color{blue}{\left(2 \cdot \frac{3}{2}\right)}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{b}^{\left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot 3\right)}\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      7. pow-sqr N/A

         \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{{b}^{\left(\frac{1}{2} \cdot 3\right)} \cdot {b}^{\left(\frac{1}{2} \cdot 3\right)}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      8. unpow-prod-down N/A

         \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{{\left(b \cdot b\right)}^{\left(\frac{1}{2} \cdot 3\right)}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      9. sqr-neg N/A

         \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      12. pow2 N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\color{blue}{\left({\left(\mathsf{neg}\left(b\right)\right)}^{2}\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      13. pow-pow N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{{\left(\mathsf{neg}\left(b\right)\right)}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      14. lift-neg.f64 N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\left(2 \cdot \color{blue}{\frac{3}{2}}\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\color{blue}{3}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      17. cube-neg N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{\mathsf{neg}\left({b}^{3}\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      18. neg-sub0 N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{0 - {b}^{3}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{{0}^{3}} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

      20. flip3-- N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \color{blue}{\left(0 - b\right)}} \]

      21. neg-sub0 N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}} \]

      22. lift-neg.f64 N/A

          \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}} \]

   7. Applied rewrites 26.2%

      \[\leadsto \frac{c}{\color{blue}{\frac{0}{\mathsf{fma}\left(b, b, 0\right)} - \left(-b\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c}{\frac{0}{\color{blue}{b \cdot b} + 0} - \left(\mathsf{neg}\left(b\right)\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{c}{\frac{0}{\color{blue}{b \cdot b}} - \left(\mathsf{neg}\left(b\right)\right)} \]

      3. div0 N/A

         \[\leadsto \frac{c}{\color{blue}{0} - \left(\mathsf{neg}\left(b\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{c}{0 - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}} \]

      5. neg-sub0 N/A

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg 26.2

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

   9. Applied rewrites 26.2%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5200000000000:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 35.2

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

5. Applied rewrites 35.2%

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

   1. neg-sub0 N/A

      \[\leadsto \frac{c}{\color{blue}{0 - b}} \]

   2. flip3-- N/A

      \[\leadsto \frac{c}{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{c}{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   4. div-sub N/A

      \[\leadsto \frac{c}{\color{blue}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{b}^{\color{blue}{\left(2 \cdot \frac{3}{2}\right)}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{b}^{\left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot 3\right)}\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   7. pow-sqr N/A

      \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{{b}^{\left(\frac{1}{2} \cdot 3\right)} \cdot {b}^{\left(\frac{1}{2} \cdot 3\right)}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   8. unpow-prod-down N/A

      \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{{\left(b \cdot b\right)}^{\left(\frac{1}{2} \cdot 3\right)}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   9. sqr-neg N/A

      \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   10. lift-neg.f64 N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   12. pow2 N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\color{blue}{\left({\left(\mathsf{neg}\left(b\right)\right)}^{2}\right)}}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   13. pow-pow N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{{\left(\mathsf{neg}\left(b\right)\right)}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   14. lift-neg.f64 N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\left(2 \cdot \color{blue}{\frac{3}{2}}\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   16. metadata-eval N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\color{blue}{3}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   17. cube-neg N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{\mathsf{neg}\left({b}^{3}\right)}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   18. neg-sub0 N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{0 - {b}^{3}}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   19. metadata-eval N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \frac{\color{blue}{{0}^{3}} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} \]

   20. flip3-- N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \color{blue}{\left(0 - b\right)}} \]

   21. neg-sub0 N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}} \]

   22. lift-neg.f64 N/A

       \[\leadsto \frac{c}{\frac{0}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}} \]

7. Applied rewrites 9.7%

   \[\leadsto \frac{c}{\color{blue}{\frac{0}{\mathsf{fma}\left(b, b, 0\right)} - \left(-b\right)}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{c}{\frac{0}{\color{blue}{b \cdot b} + 0} - \left(\mathsf{neg}\left(b\right)\right)} \]

   2. +-rgt-identity N/A

      \[\leadsto \frac{c}{\frac{0}{\color{blue}{b \cdot b}} - \left(\mathsf{neg}\left(b\right)\right)} \]

   3. div0 N/A

      \[\leadsto \frac{c}{\color{blue}{0} - \left(\mathsf{neg}\left(b\right)\right)} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{c}{0 - \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}} \]

   5. neg-sub0 N/A

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

   6. lift-neg.f64 N/A

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

   7. remove-double-neg 10.2

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

9. Applied rewrites 10.2%

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

Alternative 13: 2.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 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 = b / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 30.5

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

5. Applied rewrites 30.5%

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

   1. lift-neg.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(a\right)} \cdot b} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(a\right)} \cdot b} \]

   5. metadata-eval N/A

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

   6. lift-neg.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 30.4

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

7. Applied rewrites 30.4%

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

   1. associate-*l/ N/A

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

9. Applied rewrites 2.7%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
10. 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 2024216 
(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)))