quadp (p42, positive)

Percentage Accurate: 52.5% → 85.9%

Time: 9.6s

Alternatives: 8

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: 52.5% 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: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a}, 0.5, -0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e+137)
   (/ (- b) a)
   (if (<= b 4e-55)
     (fma (/ (sqrt (fma (* c a) -4.0 (* b b))) a) 0.5 (* -0.5 (/ b a)))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e+137) {
		tmp = -b / a;
	} else if (b <= 4e-55) {
		tmp = fma((sqrt(fma((c * a), -4.0, (b * b))) / a), 0.5, (-0.5 * (b / a)));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e+137)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4e-55)
		tmp = fma(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) / a), 0.5, Float64(-0.5 * Float64(b / a)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e+137], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4e-55], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] * 0.5 + N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.79999999999999963e137

   1. Initial program 39.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if -3.79999999999999963e137 < b < 3.99999999999999998e-55

   1. Initial program 82.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 82.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 82.5

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

   4. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. div-inv N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. times-frac N/A

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

      18. metadata-eval N/A

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

   6. Applied rewrites 82.8%

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

   7. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.8

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

   9. Applied rewrites 82.8%

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

   if 3.99999999999999998e-55 < b

   1. Initial program 13.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a}, 0.5, -0.5 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+134}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e+134)
   (/ (- b) a)
   (if (<= b 4e-55)
     (* (- (sqrt (fma -4.0 (* c a) (* b b))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e+134) {
		tmp = -b / a;
	} else if (b <= 4e-55) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.85e+134)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4e-55)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e+134], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4e-55], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.85000000000000007e134

   1. Initial program 41.7%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -1.85000000000000007e134 < b < 3.99999999999999998e-55

   1. Initial program 81.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 82.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 82.3

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

   4. Applied rewrites 82.3%

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

   if 3.99999999999999998e-55 < b

   1. Initial program 13.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+134}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{\left(\sqrt{-4 \cdot \left(c \cdot a\right)} - b\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.6e-88)
   (fma (/ c (* b b)) b (/ (- b) a))
   (if (<= b 2.1e-55)
     (/ (* (- (sqrt (* -4.0 (* c a))) b) 0.5) a)
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.6e-88) {
		tmp = fma((c / (b * b)), b, (-b / a));
	} else if (b <= 2.1e-55) {
		tmp = ((sqrt((-4.0 * (c * a))) - b) * 0.5) / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.6e-88)
		tmp = fma(Float64(c / Float64(b * b)), b, Float64(Float64(-b) / a));
	elseif (b <= 2.1e-55)
		tmp = Float64(Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) * 0.5) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.6e-88], N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[((-b) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-55], N[(N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.5), $MachinePrecision] / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5999999999999998e-88

   1. Initial program 67.2%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   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. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -9.5999999999999998e-88 < b < 2.1000000000000002e-55

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 75.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 72.4

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

   7. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 72.6

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

   9. Applied rewrites 72.6%

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

   if 2.1000000000000002e-55 < b

   1. Initial program 13.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{\left(\sqrt{-4 \cdot \left(c \cdot a\right)} - b\right) \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.6e-88)
   (fma (/ c (* b b)) b (/ (- b) a))
   (if (<= b 2.1e-55)
     (* (- (sqrt (* -4.0 (* c a))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.6e-88) {
		tmp = fma((c / (b * b)), b, (-b / a));
	} else if (b <= 2.1e-55) {
		tmp = (sqrt((-4.0 * (c * a))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.6e-88)
		tmp = fma(Float64(c / Float64(b * b)), b, Float64(Float64(-b) / a));
	elseif (b <= 2.1e-55)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.6e-88], N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[((-b) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-55], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5999999999999998e-88

   1. Initial program 67.2%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   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. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -9.5999999999999998e-88 < b < 2.1000000000000002e-55

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 75.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 75.8

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if 2.1000000000000002e-55 < b

   1. Initial program 13.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.6 \cdot 10^{-88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (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 <= (-2d-310)) then
        tmp = (c / b) - (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 <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(c / b) - 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 <= -2e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 74.0%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   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. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 62.3

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

   5. Applied rewrites 62.3%

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

   7. Applied rewrites 62.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 29.6%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 65.7

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

   5. Applied rewrites 65.7%

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

Alternative 6: 67.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-239}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.5e-239) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5e-239) {
		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 <= 9.5d-239) 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 <= 9.5e-239) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.5e-239:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.5e-239)
		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 <= 9.5e-239)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.5e-239], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.4999999999999992e-239

   1. Initial program 74.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if 9.4999999999999992e-239 < b

   1. Initial program 22.4%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 73.8

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

   5. Applied rewrites 73.8%

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

Alternative 7: 43.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{-125}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5.4e-125) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.4e-125) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.4d-125) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.4e-125) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.4e-125:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.4e-125)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.4e-125)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.4e-125], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 5.3999999999999995e-125

   1. Initial program 71.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 52.9

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

   5. Applied rewrites 52.9%

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

   if 5.3999999999999995e-125 < b

   1. Initial program 20.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 20.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 20.0

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

   4. Applied rewrites 20.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. div-inv N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. times-frac N/A

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

      18. metadata-eval N/A

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

   6. Applied rewrites 16.8%

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

   7. Taylor expanded in c around 0

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

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

      3. mul0-rgt 27.0

         \[\leadsto \color{blue}{0} \]

   9. Applied rewrites 27.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 10.9% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 54.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 54.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 54.8

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

4. Applied rewrites 54.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. lift--.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. div-inv N/A

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

   14. *-rgt-identity N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. times-frac N/A

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

   18. metadata-eval N/A

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

6. Applied rewrites 53.9%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

   3. mul0-rgt 10.8

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 10.8%

   \[\leadsto \color{blue}{0} \]
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 2024240 
(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)))