Quadratic roots, full range

Percentage Accurate: 52.1% → 83.9%

Time: 7.7s

Alternatives: 6

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: 52.1% 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: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+77}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 4700000:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e+77)
   (- (fma (/ (- c) (* b b)) b (/ b a)))
   (if (<= b 4700000.0)
     (/ (+ (- b) (sqrt (fma (* c -4.0) a (* b b)))) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e+77) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 4700000.0) {
		tmp = (-b + sqrt(fma((c * -4.0), a, (b * b)))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e+77)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 4700000.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e+77], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 4700000.0], N[(N[((-b) + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.39999999999999997e77

   1. Initial program 61.8%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   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. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if -3.39999999999999997e77 < b < 4.7e6

   1. Initial program 82.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 82.9

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

   4. Applied rewrites 82.9%

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

   if 4.7e6 < b

   1. Initial program 10.1%

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

   3. Taylor expanded in a 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. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 87.3

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

   5. Applied rewrites 87.3%

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

Alternative 2: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-20}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-20)
   (- (fma (/ (- c) (* b b)) b (/ b a)))
   (if (<= b 9e-13)
     (/ (+ (- b) (sqrt (* (* c a) -4.0))) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-20) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 9e-13) {
		tmp = (-b + sqrt(((c * a) * -4.0))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e-20)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 9e-13)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(c * a) * -4.0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e-20], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 9e-13], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.99999999999999956e-20

   1. Initial program 70.0%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   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. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if -7.99999999999999956e-20 < b < 9e-13

   1. Initial program 80.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   if 9e-13 < b

   1. Initial program 11.2%

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

   3. Taylor expanded in a 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. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 86.5

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

   5. Applied rewrites 86.5%

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

Alternative 3: 66.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- (fma (/ (- c) (* b b)) b (/ b a))) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 74.7%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   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. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if -3.999999999999988e-310 < 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. Taylor expanded in a 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. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

Alternative 4: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-305}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 7e-305) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7e-305) {
		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 <= 7d-305) 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 <= 7e-305) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7e-305:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7e-305)
		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 <= 7e-305)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7e-305], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.9999999999999996e-305

   1. Initial program 74.8%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.2

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

   5. Applied rewrites 67.2%

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

   if 6.9999999999999996e-305 < b

   1. Initial program 33.6%

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

   3. Taylor expanded in a 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. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 63.2

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

   5. Applied rewrites 63.2%

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

Alternative 5: 44.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-302}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -3.3e-302) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-302) {
		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 <= (-3.3d-302)) 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 <= -3.3e-302) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.3e-302:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-302)
		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 <= -3.3e-302)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-302], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -3.3000000000000002e-302

   1. Initial program 75.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if -3.3000000000000002e-302 < b

   1. Initial program 33.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 33.9

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

   4. Applied rewrites 33.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-neg.f64 N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

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

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

      19. metadata-eval N/A

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

      20. lower-*.f64 33.6

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. metadata-eval N/A

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

      6. +-inverses 14.6

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

   9. Applied rewrites 14.6%

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

Alternative 6: 11.3% 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 54.5%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 54.6

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

4. Applied rewrites 54.6%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. frac-2neg N/A

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

   14. lift-neg.f64 N/A

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

   15. remove-double-neg N/A

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

   16. lower-/.f64 N/A

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

   17. lift-*.f64 N/A

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

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

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

   19. metadata-eval N/A

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

   20. lower-*.f64 54.4

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

6. Applied rewrites 54.4%

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

7. Taylor expanded in a around 0

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

   1. div-add N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r/ N/A

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

   4. fp-cancel-sign-sub-inv N/A

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

   5. metadata-eval N/A

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

   6. +-inverses 8.6

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

9. Applied rewrites 8.6%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))