Quadratic roots, full range

Percentage Accurate: 50.7% → 84.6%

Time: 10.7s

Alternatives: 8

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+168)
   (/ b (- a))
   (if (<= b 1.2e-103)
     (/ (- (sqrt (fma b b (* a (* c -4.0)))) b) (* a 2.0))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+168) {
		tmp = b / -a;
	} else if (b <= 1.2e-103) {
		tmp = (sqrt(fma(b, b, (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+168)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.2e-103)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+168], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.2e-103], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999983e168

   1. Initial program 42.3%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.49999999999999983e168 < b < 1.2000000000000001e-103

   1. Initial program 87.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 87.8

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

   4. Applied rewrites 87.8%

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

   if 1.2000000000000001e-103 < b

   1. Initial program 21.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+115}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-107}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e+115)
   (/ b (- a))
   (if (<= b 4.9e-107)
     (* (- (sqrt (fma b b (* a (* c -4.0)))) b) (/ 0.5 a))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e+115) {
		tmp = b / -a;
	} else if (b <= 4.9e-107) {
		tmp = (sqrt(fma(b, b, (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e+115)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4.9e-107)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e+115], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4.9e-107], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $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 < -1.9e115

   1. Initial program 57.7%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.9e115 < b < 4.8999999999999998e-107

   1. Initial program 85.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 86.0

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

   4. Applied rewrites 86.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 85.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 85.8

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 85.6

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

   6. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 85.8

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

   8. Applied rewrites 85.8%

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

   if 4.8999999999999998e-107 < b

   1. Initial program 21.9%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+115}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-107}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+115}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e+115)
   (/ b (- a))
   (if (<= b 6.5e-126)
     (* (/ -0.5 a) (- b (sqrt (fma c (* a -4.0) (* b b)))))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e+115) {
		tmp = b / -a;
	} else if (b <= 6.5e-126) {
		tmp = (-0.5 / a) * (b - sqrt(fma(c, (a * -4.0), (b * b))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e+115)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.5e-126)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e+115], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.5e-126], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9e115

   1. Initial program 57.7%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.9e115 < b < 6.50000000000000014e-126

   1. Initial program 87.2%

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

   4. Applied rewrites 87.0%

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

   if 6.50000000000000014e-126 < b

   1. Initial program 23.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+115}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e-41)
   (/ b (- a))
   (if (<= b 6.6e-127)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e-41) {
		tmp = b / -a;
	} else if (b <= 6.6e-127) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} 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 <= (-2.25d-41)) then
        tmp = b / -a
    else if (b <= 6.6d-127) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    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 <= -2.25e-41) {
		tmp = b / -a;
	} else if (b <= 6.6e-127) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.25e-41:
		tmp = b / -a
	elif b <= 6.6e-127:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.25e-41)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.6e-127)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.25e-41)
		tmp = b / -a;
	elseif (b <= 6.6e-127)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.25e-41], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.6e-127], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -2.25e-41

   1. Initial program 72.1%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   if -2.25e-41 < b < 6.59999999999999961e-127

   1. Initial program 82.0%

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

   3. Taylor expanded in b around 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 73.3

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

   5. Applied rewrites 73.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 73.3

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

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{a \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(-4 \cdot c\right)\right)} - b}{\color{blue}{2 \cdot a}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sqrt{a \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(-4 \cdot c\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{a \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(-4 \cdot c\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

   7. Applied rewrites 73.3%

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

   if 6.59999999999999961e-127 < b

   1. Initial program 23.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-127}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e-41)
   (/ b (- a))
   (if (<= b 6.6e-127)
     (* (/ 0.5 a) (- (sqrt (* a (* c -4.0))) b))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e-41) {
		tmp = b / -a;
	} else if (b <= 6.6e-127) {
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	} 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 <= (-2.25d-41)) then
        tmp = b / -a
    else if (b <= 6.6d-127) then
        tmp = (0.5d0 / a) * (sqrt((a * (c * (-4.0d0)))) - b)
    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 <= -2.25e-41) {
		tmp = b / -a;
	} else if (b <= 6.6e-127) {
		tmp = (0.5 / a) * (Math.sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.25e-41:
		tmp = b / -a
	elif b <= 6.6e-127:
		tmp = (0.5 / a) * (math.sqrt((a * (c * -4.0))) - b)
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.25e-41)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.6e-127)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(a * Float64(c * -4.0))) - b));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.25e-41)
		tmp = b / -a;
	elseif (b <= 6.6e-127)
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.25e-41], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.6e-127], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -2.25e-41

   1. Initial program 72.1%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 95.7

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

   5. Applied rewrites 95.7%

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

   if -2.25e-41 < b < 6.59999999999999961e-127

   1. Initial program 82.0%

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

   3. Taylor expanded in b around 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 73.3

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

   5. Applied rewrites 73.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 73.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if 6.59999999999999961e-127 < b

   1. Initial program 23.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-127}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-298}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.32e-298) (/ b (- a)) (- (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.32e-298) {
		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 <= 1.32d-298) 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 <= 1.32e-298) {
		tmp = b / -a;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.32e-298:
		tmp = b / -a
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.32e-298)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.32e-298)
		tmp = b / -a;
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.32e-298], N[(b / (-a)), $MachinePrecision], (-N[(c / b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 1.3200000000000001e-298

   1. Initial program 78.0%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   if 1.3200000000000001e-298 < b

   1. Initial program 32.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 70.6

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

   5. Applied rewrites 70.6%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-298}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -1e-309) (/ b (- a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-309:
		tmp = b / -a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-309)
		tmp = Float64(b / Float64(-a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-309)
		tmp = b / -a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-309], N[(b / (-a)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.000000000000002e-309

   1. Initial program 78.0%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 70.8

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

   5. Applied rewrites 70.8%

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

   if -1.000000000000002e-309 < b

   1. Initial program 34.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 34.2

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 34.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 34.2

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 34.1

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

   6. Applied rewrites 34.1%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 30.3%

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

   9. Taylor expanded in a around 0

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

       1. distribute-rgt-out N/A

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

       2. metadata-eval N/A

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

       3. associate-*l/ N/A

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

       4. mul0-rgt 19.2

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

   11. Applied rewrites 19.2%

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

Alternative 8: 11.2% 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 55.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

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

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 55.8

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

4. Applied rewrites 55.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 55.7

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 55.7

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

   14. lift-fma.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-fma.f64 55.6

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

6. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. distribute-lft-in N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. associate-/r* N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

8. Applied rewrites 53.7%

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

9. Taylor expanded in a around 0

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

    1. distribute-rgt-out N/A

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

    2. metadata-eval N/A

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

    3. associate-*l/ N/A

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

    4. mul0-rgt 11.0

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

11. Applied rewrites 11.0%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))