Quadratic roots, narrow range

Percentage Accurate: 56.0% → 99.6%

Time: 24.2s

Alternatives: 6

Speedup: 29.0×

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: 56.0% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in a around inf 50.6%

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

   1. flip-+ 50.5%

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

   2. pow2 50.5%

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

   3. add-sqr-sqrt 51.7%

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

   4. cancel-sign-sub-inv 51.7%

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

   5. metadata-eval 51.7%

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

   6. cancel-sign-sub-inv 51.7%

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

   7. metadata-eval 51.7%

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

7. Applied egg-rr 51.7%

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

8. Taylor expanded in b around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

10. Simplified 99.3%

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

    1. div-inv 99.2%

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

    2. associate-/l* 99.3%

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

    3. +-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. fma-define 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. associate-*r/ 99.2%

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

    3. associate-*r* 99.2%

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

    4. times-frac 99.3%

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

    5. *-lft-identity 99.3%

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

    6. associate-/r* 99.5%

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

    7. associate-*r* 99.5%

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

    8. times-frac 99.5%

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

    9. *-commutative 99.5%

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

    10. metadata-eval 99.5%

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

14. Simplified 99.5%

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

15. Taylor expanded in a around 0 99.5%

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

16. Taylor expanded in a around 0 99.6%

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

17. Final simplification 99.6%

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

Alternative 2: 85.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -4

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

      2. +-commutative 85.8%

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

      3. sqr-neg 85.8%

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

      4. unsub-neg 85.8%

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

      5. sqr-neg 85.8%

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

      6. fma-neg 85.9%

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

      7. distribute-lft-neg-in 85.9%

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

      8. *-commutative 85.9%

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

      9. *-commutative 85.9%

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

      10. distribute-rgt-neg-in 85.9%

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

      11. metadata-eval 85.9%

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

   3. Simplified 85.9%

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 47.0%

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

      1. *-commutative 47.0%

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

   3. Simplified 47.0%

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

   5. Taylor expanded in a around inf 46.6%

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

      1. flip-+ 46.4%

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

      2. pow2 46.4%

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

      3. add-sqr-sqrt 47.7%

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

      4. cancel-sign-sub-inv 47.7%

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

      5. metadata-eval 47.7%

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

      6. cancel-sign-sub-inv 47.7%

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

      7. metadata-eval 47.7%

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

   7. Applied egg-rr 47.7%

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

   8. Taylor expanded in b around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. *-commutative 99.3%

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

   10. Simplified 99.3%

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

       1. div-inv 99.2%

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

       2. associate-/l* 99.3%

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

       3. +-commutative 99.3%

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

       4. *-commutative 99.3%

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

       5. fma-define 99.3%

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

   12. Applied egg-rr 99.3%

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

       1. *-commutative 99.3%

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

       2. associate-*r/ 99.2%

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

       3. associate-*r* 99.2%

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

       4. times-frac 99.3%

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

       5. *-lft-identity 99.3%

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

       6. associate-/r* 99.5%

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

       7. associate-*r* 99.5%

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

       8. times-frac 99.5%

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

       9. *-commutative 99.5%

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

       10. metadata-eval 99.5%

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

   14. Simplified 99.5%

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

   15. Taylor expanded in a around 0 99.5%

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

   16. Taylor expanded in a around 0 88.5%

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

       1. distribute-lft-out-- 88.5%

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

       2. *-commutative 88.5%

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

   18. Simplified 88.5%

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

4. Final simplification 88.2%

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

Alternative 3: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -4:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{a}}{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -4.0)
     t_0
     (/ (* 2.0 (/ (* c a) a)) (* 2.0 (- (/ (* c a) b) b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((c * a) / a)) / (2.0 * (((c * a) / b) - 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    if (t_0 <= (-4.0d0)) then
        tmp = t_0
    else
        tmp = (2.0d0 * ((c * a) / a)) / (2.0d0 * (((c * a) / b) - b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((c * a) / a)) / (2.0 * (((c * a) / b) - b));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	tmp = 0
	if t_0 <= -4.0:
		tmp = t_0
	else:
		tmp = (2.0 * ((c * a) / a)) / (2.0 * (((c * a) / b) - b))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a))
	tmp = 0.0
	if (t_0 <= -4.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(c * a) / a)) / Float64(2.0 * Float64(Float64(Float64(c * a) / b) - b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	tmp = 0.0;
	if (t_0 <= -4.0)
		tmp = t_0;
	else
		tmp = (2.0 * ((c * a) / a)) / (2.0 * (((c * a) / b) - b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4.0], t$95$0, N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -4

   1. Initial program 85.8%

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 47.0%

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

      1. *-commutative 47.0%

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

   3. Simplified 47.0%

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

   5. Taylor expanded in a around inf 46.6%

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

      1. flip-+ 46.4%

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

      2. pow2 46.4%

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

      3. add-sqr-sqrt 47.7%

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

      4. cancel-sign-sub-inv 47.7%

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

      5. metadata-eval 47.7%

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

      6. cancel-sign-sub-inv 47.7%

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

      7. metadata-eval 47.7%

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

   7. Applied egg-rr 47.7%

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

   8. Taylor expanded in b around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. *-commutative 99.3%

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

   10. Simplified 99.3%

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

       1. div-inv 99.2%

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

       2. associate-/l* 99.3%

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

       3. +-commutative 99.3%

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

       4. *-commutative 99.3%

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

       5. fma-define 99.3%

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

   12. Applied egg-rr 99.3%

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

       1. *-commutative 99.3%

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

       2. associate-*r/ 99.2%

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

       3. associate-*r* 99.2%

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

       4. times-frac 99.3%

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

       5. *-lft-identity 99.3%

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

       6. associate-/r* 99.5%

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

       7. associate-*r* 99.5%

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

       8. times-frac 99.5%

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

       9. *-commutative 99.5%

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

       10. metadata-eval 99.5%

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

   14. Simplified 99.5%

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

   15. Taylor expanded in a around 0 99.5%

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

   16. Taylor expanded in a around 0 88.5%

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

       1. distribute-lft-out-- 88.5%

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

       2. *-commutative 88.5%

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

   18. Simplified 88.5%

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

4. Final simplification 88.2%

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

Alternative 4: 82.1% accurate, 6.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (2.0 * ((c * a) / a)) / (2.0 * (((c * a) / b) - 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 = (2.0d0 * ((c * a) / a)) / (2.0d0 * (((c * a) / b) - b))
end function

Java

public static double code(double a, double b, double c) {
	return (2.0 * ((c * a) / a)) / (2.0 * (((c * a) / b) - b));
}

Python

def code(a, b, c):
	return (2.0 * ((c * a) / a)) / (2.0 * (((c * a) / b) - b))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in a around inf 50.6%

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

   1. flip-+ 50.5%

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

   2. pow2 50.5%

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

   3. add-sqr-sqrt 51.7%

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

   4. cancel-sign-sub-inv 51.7%

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

   5. metadata-eval 51.7%

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

   6. cancel-sign-sub-inv 51.7%

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

   7. metadata-eval 51.7%

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

7. Applied egg-rr 51.7%

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

8. Taylor expanded in b around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

10. Simplified 99.3%

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

    1. div-inv 99.2%

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

    2. associate-/l* 99.3%

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

    3. +-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. fma-define 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. associate-*r/ 99.2%

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

    3. associate-*r* 99.2%

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

    4. times-frac 99.3%

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

    5. *-lft-identity 99.3%

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

    6. associate-/r* 99.5%

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

    7. associate-*r* 99.5%

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

    8. times-frac 99.5%

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

    9. *-commutative 99.5%

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

    10. metadata-eval 99.5%

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

14. Simplified 99.5%

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

15. Taylor expanded in a around 0 99.5%

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

16. Taylor expanded in a around 0 85.0%

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

    1. distribute-lft-out-- 85.0%

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

    2. *-commutative 85.0%

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

18. Simplified 85.0%

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

19. Final simplification 85.0%

    \[\leadsto \frac{2 \cdot \frac{c \cdot a}{a}}{2 \cdot \left(\frac{c \cdot a}{b} - b\right)} \]
20. Add Preprocessing

Alternative 5: 64.0% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(c / Float64(-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 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in b around inf 68.6%

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

   1. associate-*r/ 68.6%

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

   2. mul-1-neg 68.6%

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

7. Simplified 68.6%

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

8. Final simplification 68.6%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Alternative 6: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. *-commutative 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in b around inf 50.9%

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

   1. expm1-log1p-u 40.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{{b}^{2} \cdot \left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{a \cdot 2}\right)\right)} \]

   2. expm1-undefine 38.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{{b}^{2} \cdot \left(1 + -4 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{a \cdot 2}\right)} - 1} \]

7. Applied egg-rr 38.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} \cdot \mathsf{fma}\left(-4, \left(c \cdot a\right) \cdot {b}^{-2}, 1\right)}\right)}{a \cdot 2}\right)} - 1} \]

8. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

10. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))