Quadratic roots, narrow range

Percentage Accurate: 55.5% → 99.3%

Time: 8.7s

Alternatives: 9

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.5% 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: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

4. Applied rewrites 54.9%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

6. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot c\right) \cdot a + 0}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)} \]

   5. +-rgt-identity N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 102:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -4, a, 0\right)}{\mathsf{fma}\left(4, b, \left(\frac{c}{b} \cdot a\right) \cdot -4\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 102.0)
   (/ (- (sqrt (fma b b (* (* c -4.0) a))) b) (* 2.0 a))
   (/ (fma (* c -4.0) a 0.0) (* (fma 4.0 b (* (* (/ c b) a) -4.0)) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 102.0) {
		tmp = (sqrt(fma(b, b, ((c * -4.0) * a))) - b) / (2.0 * a);
	} else {
		tmp = fma((c * -4.0), a, 0.0) / (fma(4.0, b, (((c / b) * a) * -4.0)) * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 102.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -4.0) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c * -4.0), a, 0.0) / Float64(fma(4.0, b, Float64(Float64(Float64(c / b) * a) * -4.0)) * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 102.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -4.0), $MachinePrecision] * a + 0.0), $MachinePrecision] / N[(N[(4.0 * b + N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 102

   1. Initial program 81.1%

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-+.f64 N/A

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

   6. Applied rewrites 80.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. unsub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. un-div-inv N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l/ N/A

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

      18. metadata-eval N/A

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

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

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

      20. neg-mul-1 N/A

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

      21. remove-double-neg N/A

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

   8. Applied rewrites 81.3%

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

   if 102 < b

   1. Initial program 44.5%

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

   4. Applied rewrites 43.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

   6. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\color{blue}{a \cdot \left(-4 \cdot \frac{a \cdot c}{b} + 4 \cdot b\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\color{blue}{a \cdot \left(-4 \cdot \frac{a \cdot c}{b} + 4 \cdot b\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{a \cdot \color{blue}{\left(4 \cdot b + -4 \cdot \frac{a \cdot c}{b}\right)}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{a \cdot \color{blue}{\mathsf{fma}\left(4, b, -4 \cdot \frac{a \cdot c}{b}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{a \cdot \mathsf{fma}\left(4, b, \color{blue}{-4 \cdot \frac{a \cdot c}{b}}\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{a \cdot \mathsf{fma}\left(4, b, -4 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{a \cdot \mathsf{fma}\left(4, b, -4 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)} \]

      7. lower-/.f64 92.6

         \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{a \cdot \mathsf{fma}\left(4, b, -4 \cdot \left(a \cdot \color{blue}{\frac{c}{b}}\right)\right)} \]

   9. Applied rewrites 92.6%

      \[\leadsto \frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\color{blue}{a \cdot \mathsf{fma}\left(4, b, -4 \cdot \left(a \cdot \frac{c}{b}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 102:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -4, a, 0\right)}{\mathsf{fma}\left(4, b, \left(\frac{c}{b} \cdot a\right) \cdot -4\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

4. Applied rewrites 54.9%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

6. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, 0\right) \cdot 1}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot c\right) \cdot a + 0}}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)} \]

   5. +-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 99.3

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.3

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.3

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 4: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 102:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 102.0)
   (/ (- (sqrt (fma b b (* (* c -4.0) a))) b) (* 2.0 a))
   (/ (fma (* c c) (/ a (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 102.0) {
		tmp = (sqrt(fma(b, b, ((c * -4.0) * a))) - b) / (2.0 * a);
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 102.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -4.0) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 102.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 102

   1. Initial program 81.1%

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. lower-+.f64 N/A

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

   6. Applied rewrites 80.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. unsub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. un-div-inv N/A

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

      13. lift-/.f64 N/A

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

      14. metadata-eval N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l/ N/A

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

      18. metadata-eval N/A

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

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

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

      20. neg-mul-1 N/A

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

      21. remove-double-neg N/A

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

   8. Applied rewrites 81.3%

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

   if 102 < b

   1. Initial program 44.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 92.5

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

   8. Applied rewrites 92.5%

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

   10. Applied rewrites 92.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 102:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 102:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 102.0)
   (* (- (sqrt (fma (* c -4.0) a (* b b))) b) (/ 0.5 a))
   (/ (fma (* c c) (/ a (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 102.0) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 102.0)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 102.0], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 102

   1. Initial program 81.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 81.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 81.2

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

   4. Applied rewrites 81.2%

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

   if 102 < b

   1. Initial program 44.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 92.5

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

   8. Applied rewrites 92.5%

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

   10. Applied rewrites 92.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 102:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

5. Applied rewrites 91.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-neg.f64 82.9

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

8. Applied rewrites 82.9%

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

10. Applied rewrites 82.9%

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

Alternative 7: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* (fma (- a) (/ c (* b b)) -1.0) c) b))

C

double code(double a, double b, double c) {
	return (fma(-a, (c / (b * b)), -1.0) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(-a), Float64(c / Float64(b * b)), -1.0) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[((-a) * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

5. Applied rewrites 88.7%

   \[\leadsto \color{blue}{\frac{\frac{\left(\left({c}^{3} \cdot a\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 82.8%

   \[\leadsto \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right) \cdot c}{b} \]
9. Add Preprocessing

Alternative 8: 64.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 64.8

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

5. Applied rewrites 64.8%

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

Alternative 9: 3.2% 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 55.3%

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

4. Applied rewrites 54.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f64 N/A

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

6. Applied rewrites 54.7%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.2

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

9. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))