Quadratic roots, narrow range

Percentage Accurate: 55.4% → 99.2%

Time: 16.9s

Alternatives: 8

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: 55.4% 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: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -4\right)\\ \frac{t\_0}{b + \sqrt{t\_0 + {b}^{2}}} \cdot \frac{-1}{a \cdot -2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -4.0))))
   (* (/ t_0 (+ b (sqrt (+ t_0 (pow b 2.0))))) (/ -1.0 (* a -2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * -4.0);
	return (t_0 / (b + sqrt((t_0 + pow(b, 2.0))))) * (-1.0 / (a * -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
    real(8) :: t_0
    t_0 = a * (c * (-4.0d0))
    code = (t_0 / (b + sqrt((t_0 + (b ** 2.0d0))))) * ((-1.0d0) / (a * (-2.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * -4.0);
	return (t_0 / (b + Math.sqrt((t_0 + Math.pow(b, 2.0))))) * (-1.0 / (a * -2.0));
}

Python

def code(a, b, c):
	t_0 = a * (c * -4.0)
	return (t_0 / (b + math.sqrt((t_0 + math.pow(b, 2.0))))) * (-1.0 / (a * -2.0))

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -4.0))
	return Float64(Float64(t_0 / Float64(b + sqrt(Float64(t_0 + (b ^ 2.0))))) * Float64(-1.0 / Float64(a * -2.0)))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * -4.0);
	tmp = (t_0 / (b + sqrt((t_0 + (b ^ 2.0))))) * (-1.0 / (a * -2.0));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(b + N[Sqrt[N[(t$95$0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 51.1%

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

   1. +-commutative 51.1%

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

   2. sqr-neg 51.1%

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

   3. unsub-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sub-neg 51.1%

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

   6. +-commutative 51.1%

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

   7. *-commutative 51.1%

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

   8. associate-*r* 51.1%

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

   9. distribute-rgt-neg-in 51.1%

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

   10. fma-define 51.1%

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

   11. *-commutative 51.1%

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

   12. distribute-rgt-neg-in 51.1%

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

   13. metadata-eval 51.1%

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

3. Simplified 51.1%

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

   1. frac-2neg 51.1%

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

   2. div-inv 51.1%

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

   3. sub-neg 51.1%

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

   4. distribute-neg-in 51.1%

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

   5. pow2 51.1%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 51.1%

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

   13. sqr-neg 51.1%

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

   14. sqrt-prod 50.2%

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

   15. add-sqr-sqrt 51.1%

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

   16. distribute-rgt-neg-in 51.1%

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

   17. metadata-eval 51.1%

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

6. Applied egg-rr 51.1%

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

   1. flip-+ 51.2%

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

   2. pow2 51.2%

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

   3. unpow2 51.2%

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

8. Applied egg-rr 51.2%

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

   1. unpow2 51.2%

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

   2. sqr-neg 51.2%

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

   3. rem-square-sqrt 52.3%

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

10. Simplified 52.3%

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

11. Taylor expanded in a around 0 99.3%

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

    1. associate-*r* 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. *-commutative 99.3%

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

13. Simplified 99.3%

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

    1. fma-undefine 99.3%

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

15. Applied egg-rr 99.3%

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

16. Final simplification 99.3%

    \[\leadsto \frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{a \cdot \left(c \cdot -4\right) + {b}^{2}}} \cdot \frac{-1}{a \cdot -2} \]
17. Add Preprocessing

Alternative 2: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{b \cdot \left(2 \cdot \frac{a \cdot c}{{b}^{2}} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.4)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (*
    (/ 1.0 (* a -2.0))
    (/ (* a (* c -4.0)) (* b (- (* 2.0 (/ (* a c) (pow b 2.0))) 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.4) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (b * ((2.0 * ((a * c) / pow(b, 2.0))) - 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(Float64(a * Float64(c * -4.0)) / Float64(b * Float64(Float64(2.0 * Float64(Float64(a * c) / (b ^ 2.0))) - 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.4], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.4000000000000004

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. +-commutative 81.8%

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

      3. sqr-neg 81.8%

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

      4. unsub-neg 81.8%

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

      5. sqr-neg 81.8%

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

      6. fma-neg 82.0%

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

      7. distribute-lft-neg-in 82.0%

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

      8. *-commutative 82.0%

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

      9. *-commutative 82.0%

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

      10. distribute-rgt-neg-in 82.0%

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

      11. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

   if 5.4000000000000004 < b

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

      2. sqr-neg 45.4%

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

      3. unsub-neg 45.4%

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

      4. sqr-neg 45.4%

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

      5. sub-neg 45.4%

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

      6. +-commutative 45.4%

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

      7. *-commutative 45.4%

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

      8. associate-*r* 45.4%

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

      9. distribute-rgt-neg-in 45.4%

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

      10. fma-define 45.5%

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

      11. *-commutative 45.5%

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

      12. distribute-rgt-neg-in 45.5%

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

      13. metadata-eval 45.5%

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

   3. Simplified 45.5%

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

      1. frac-2neg 45.5%

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

      2. div-inv 45.4%

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

      3. sub-neg 45.4%

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

      4. distribute-neg-in 45.4%

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

      5. pow2 45.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 45.4%

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

      13. sqr-neg 45.4%

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

      14. sqrt-prod 44.7%

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

      15. add-sqr-sqrt 45.4%

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

      16. distribute-rgt-neg-in 45.4%

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

      17. metadata-eval 45.4%

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

   6. Applied egg-rr 45.4%

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

      1. flip-+ 45.5%

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

      2. pow2 45.5%

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

      3. unpow2 45.5%

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

   8. Applied egg-rr 45.5%

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

      1. unpow2 45.5%

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

      2. sqr-neg 45.5%

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

      3. rem-square-sqrt 46.6%

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

   10. Simplified 46.6%

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

   11. Taylor expanded in a around 0 99.3%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. associate-*l* 99.3%

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

       4. *-commutative 99.3%

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

   13. Simplified 99.3%

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

   14. Taylor expanded in b around inf 88.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{b \cdot \left(2 \cdot \frac{a \cdot c}{{b}^{2}} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{b \cdot \left(2 \cdot \frac{a \cdot c}{{b}^{2}} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.4)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
   (*
    (/ 1.0 (* a -2.0))
    (/ (* a (* c -4.0)) (* b (- (* 2.0 (/ (* a c) (pow b 2.0))) 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.4) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (b * ((2.0 * ((a * c) / pow(b, 2.0))) - 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.4)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(Float64(a * Float64(c * -4.0)) / Float64(b * Float64(Float64(2.0 * Float64(Float64(a * c) / (b ^ 2.0))) - 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.4], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.4000000000000004

   1. Initial program 81.8%

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

      1. +-commutative 81.8%

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

      2. sqr-neg 81.8%

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

      3. unsub-neg 81.8%

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

      4. sqr-neg 81.8%

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

      5. sub-neg 81.8%

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

      6. +-commutative 81.8%

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

      7. *-commutative 81.8%

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

      8. associate-*r* 81.8%

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

      9. distribute-rgt-neg-in 81.8%

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

      10. fma-define 81.8%

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

      11. *-commutative 81.8%

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

      12. distribute-rgt-neg-in 81.8%

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

      13. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

   if 5.4000000000000004 < b

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

      2. sqr-neg 45.4%

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

      3. unsub-neg 45.4%

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

      4. sqr-neg 45.4%

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

      5. sub-neg 45.4%

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

      6. +-commutative 45.4%

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

      7. *-commutative 45.4%

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

      8. associate-*r* 45.4%

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

      9. distribute-rgt-neg-in 45.4%

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

      10. fma-define 45.5%

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

      11. *-commutative 45.5%

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

      12. distribute-rgt-neg-in 45.5%

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

      13. metadata-eval 45.5%

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

   3. Simplified 45.5%

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

      1. frac-2neg 45.5%

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

      2. div-inv 45.4%

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

      3. sub-neg 45.4%

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

      4. distribute-neg-in 45.4%

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

      5. pow2 45.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 45.4%

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

      13. sqr-neg 45.4%

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

      14. sqrt-prod 44.7%

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

      15. add-sqr-sqrt 45.4%

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

      16. distribute-rgt-neg-in 45.4%

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

      17. metadata-eval 45.4%

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

   6. Applied egg-rr 45.4%

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

      1. flip-+ 45.5%

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

      2. pow2 45.5%

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

      3. unpow2 45.5%

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

   8. Applied egg-rr 45.5%

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

      1. unpow2 45.5%

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

      2. sqr-neg 45.5%

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

      3. rem-square-sqrt 46.6%

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

   10. Simplified 46.6%

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

   11. Taylor expanded in a around 0 99.3%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. associate-*l* 99.3%

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

       4. *-commutative 99.3%

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

   13. Simplified 99.3%

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

   14. Taylor expanded in b around inf 88.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{b \cdot \left(2 \cdot \frac{a \cdot c}{{b}^{2}} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{b \cdot \left(2 \cdot \frac{a \cdot c}{{b}^{2}} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.4)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (*
    (/ 1.0 (* a -2.0))
    (/ (* a (* c -4.0)) (* b (- (* 2.0 (/ (* a c) (pow b 2.0))) 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.4) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (b * ((2.0 * ((a * c) / pow(b, 2.0))) - 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.4d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (1.0d0 / (a * (-2.0d0))) * ((a * (c * (-4.0d0))) / (b * ((2.0d0 * ((a * c) / (b ** 2.0d0))) - 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.4) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (b * ((2.0 * ((a * c) / Math.pow(b, 2.0))) - 2.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.4:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (b * ((2.0 * ((a * c) / math.pow(b, 2.0))) - 2.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.4)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(Float64(a * Float64(c * -4.0)) / Float64(b * Float64(Float64(2.0 * Float64(Float64(a * c) / (b ^ 2.0))) - 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.4)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (b * ((2.0 * ((a * c) / (b ^ 2.0))) - 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.4], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.4000000000000004

   1. Initial program 81.8%

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

   if 5.4000000000000004 < b

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

      2. sqr-neg 45.4%

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

      3. unsub-neg 45.4%

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

      4. sqr-neg 45.4%

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

      5. sub-neg 45.4%

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

      6. +-commutative 45.4%

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

      7. *-commutative 45.4%

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

      8. associate-*r* 45.4%

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

      9. distribute-rgt-neg-in 45.4%

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

      10. fma-define 45.5%

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

      11. *-commutative 45.5%

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

      12. distribute-rgt-neg-in 45.5%

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

      13. metadata-eval 45.5%

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

   3. Simplified 45.5%

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

      1. frac-2neg 45.5%

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

      2. div-inv 45.4%

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

      3. sub-neg 45.4%

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

      4. distribute-neg-in 45.4%

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

      5. pow2 45.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 45.4%

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

      13. sqr-neg 45.4%

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

      14. sqrt-prod 44.7%

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

      15. add-sqr-sqrt 45.4%

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

      16. distribute-rgt-neg-in 45.4%

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

      17. metadata-eval 45.4%

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

   6. Applied egg-rr 45.4%

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

      1. flip-+ 45.5%

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

      2. pow2 45.5%

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

      3. unpow2 45.5%

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

   8. Applied egg-rr 45.5%

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

      1. unpow2 45.5%

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

      2. sqr-neg 45.5%

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

      3. rem-square-sqrt 46.6%

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

   10. Simplified 46.6%

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

   11. Taylor expanded in a around 0 99.3%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. associate-*l* 99.3%

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

       4. *-commutative 99.3%

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

   13. Simplified 99.3%

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

   14. Taylor expanded in b around inf 88.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{b \cdot \left(2 \cdot \frac{a \cdot c}{{b}^{2}} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (* (/ 1.0 (* a -2.0)) (/ (* a (* c -4.0)) (* 2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (2.0 * ((a * (c / 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) :: tmp
    if (b <= 1.6d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (1.0d0 / (a * (-2.0d0))) * ((a * (c * (-4.0d0))) / (2.0d0 * ((a * (c / b)) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (2.0 * ((a * (c / b)) - b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.6:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (2.0 * ((a * (c / b)) - b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(Float64(a * Float64(c * -4.0)) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.6)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (2.0 * ((a * (c / b)) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 83.8%

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

   if 1.6000000000000001 < b

   1. Initial program 46.6%

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

      1. +-commutative 46.6%

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

      2. sqr-neg 46.6%

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

      3. unsub-neg 46.6%

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

      4. sqr-neg 46.6%

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

      5. sub-neg 46.6%

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

      6. +-commutative 46.6%

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

      7. *-commutative 46.6%

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

      8. associate-*r* 46.6%

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

      9. distribute-rgt-neg-in 46.6%

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

      10. fma-define 46.6%

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

      11. *-commutative 46.6%

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

      12. distribute-rgt-neg-in 46.6%

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

      13. metadata-eval 46.6%

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

   3. Simplified 46.6%

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

      1. frac-2neg 46.6%

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

      2. div-inv 46.6%

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

      3. sub-neg 46.6%

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

      4. distribute-neg-in 46.6%

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

      5. pow2 46.6%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 46.6%

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

      13. sqr-neg 46.6%

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

      14. sqrt-prod 45.8%

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

      15. add-sqr-sqrt 46.6%

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

      16. distribute-rgt-neg-in 46.6%

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

      17. metadata-eval 46.6%

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

   6. Applied egg-rr 46.6%

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

      1. flip-+ 46.7%

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

      2. pow2 46.7%

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

      3. unpow2 46.7%

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

   8. Applied egg-rr 46.7%

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

      1. unpow2 46.7%

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

      2. sqr-neg 46.7%

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

      3. rem-square-sqrt 47.7%

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

   10. Simplified 47.7%

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

   11. Taylor expanded in a around 0 99.3%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. associate-*l* 99.3%

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

       4. *-commutative 99.3%

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

   13. Simplified 99.3%

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

   14. Taylor expanded in a around 0 88.2%

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

       1. distribute-lft-out-- 88.2%

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

       2. associate-*r/ 88.2%

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

       3. *-commutative 88.2%

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

   16. Simplified 88.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ 1.0 (* a -2.0)) (/ (* a (* c -4.0)) (* 2.0 (- (* a (/ c b)) b)))))

C

double code(double a, double b, double c) {
	return (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (2.0 * ((a * (c / 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 = (1.0d0 / (a * (-2.0d0))) * ((a * (c * (-4.0d0))) / (2.0d0 * ((a * (c / b)) - b)))
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (2.0 * ((a * (c / b)) - b)));
}

Python

def code(a, b, c):
	return (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (2.0 * ((a * (c / b)) - b)))

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(Float64(a * Float64(c * -4.0)) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (a * -2.0)) * ((a * (c * -4.0)) / (2.0 * ((a * (c / b)) - b)));
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

   1. +-commutative 51.1%

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

   2. sqr-neg 51.1%

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

   3. unsub-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sub-neg 51.1%

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

   6. +-commutative 51.1%

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

   7. *-commutative 51.1%

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

   8. associate-*r* 51.1%

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

   9. distribute-rgt-neg-in 51.1%

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

   10. fma-define 51.1%

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

   11. *-commutative 51.1%

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

   12. distribute-rgt-neg-in 51.1%

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

   13. metadata-eval 51.1%

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

3. Simplified 51.1%

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

   1. frac-2neg 51.1%

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

   2. div-inv 51.1%

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

   3. sub-neg 51.1%

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

   4. distribute-neg-in 51.1%

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

   5. pow2 51.1%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 51.1%

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

   13. sqr-neg 51.1%

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

   14. sqrt-prod 50.2%

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

   15. add-sqr-sqrt 51.1%

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

   16. distribute-rgt-neg-in 51.1%

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

   17. metadata-eval 51.1%

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

6. Applied egg-rr 51.1%

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

   1. flip-+ 51.2%

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

   2. pow2 51.2%

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

   3. unpow2 51.2%

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

8. Applied egg-rr 51.2%

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

   1. unpow2 51.2%

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

   2. sqr-neg 51.2%

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

   3. rem-square-sqrt 52.3%

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

10. Simplified 52.3%

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

11. Taylor expanded in a around 0 99.3%

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

    1. associate-*r* 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. *-commutative 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in a around 0 84.6%

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

    1. distribute-lft-out-- 84.6%

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

    2. associate-*r/ 84.6%

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

    3. *-commutative 84.6%

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

16. Simplified 84.6%

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

17. Final simplification 84.6%

    \[\leadsto \frac{1}{a \cdot -2} \cdot \frac{a \cdot \left(c \cdot -4\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)} \]
18. Add Preprocessing

Alternative 7: 81.7% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. +-commutative 51.1%

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

   2. sqr-neg 51.1%

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

   3. unsub-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sub-neg 51.1%

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

   6. +-commutative 51.1%

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

   7. *-commutative 51.1%

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

   8. associate-*r* 51.1%

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

   9. distribute-rgt-neg-in 51.1%

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

   10. fma-define 51.1%

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

   11. *-commutative 51.1%

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

   12. distribute-rgt-neg-in 51.1%

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

   13. metadata-eval 51.1%

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

3. Simplified 51.1%

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

5. Taylor expanded in a around 0 93.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative 93.1%

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

   2. mul-1-neg 93.1%

      \[\leadsto a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \color{blue}{\left(-\frac{c}{b}\right)} \]

   3. unsub-neg 93.1%

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) - \frac{c}{b}} \]

7. Simplified 93.1%

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

8. Taylor expanded in c around 0 93.1%

   \[\leadsto a \cdot \color{blue}{\left({c}^{2} \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5}}\right) - \frac{1}{{b}^{3}}\right)\right)} - \frac{c}{b} \]

9. Taylor expanded in b around inf 84.3%

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

    1. sub-neg 84.3%

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

    2. +-commutative 84.3%

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

    3. neg-mul-1 84.3%

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

    4. neg-mul-1 84.3%

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

    5. mul-1-neg 84.3%

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

    6. unsub-neg 84.3%

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

    7. associate-/l* 84.3%

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

    8. unpow2 84.3%

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

    9. unpow2 84.3%

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

    10. times-frac 84.3%

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

    11. unpow2 84.3%

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

11. Simplified 84.3%

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

    1. unpow2 84.3%

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

13. Applied egg-rr 84.3%

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

14. Final simplification 84.3%

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

Alternative 8: 64.4% 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 51.1%

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

   1. +-commutative 51.1%

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

   2. sqr-neg 51.1%

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

   3. unsub-neg 51.1%

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

   4. sqr-neg 51.1%

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

   5. sub-neg 51.1%

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

   6. +-commutative 51.1%

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

   7. *-commutative 51.1%

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

   8. associate-*r* 51.1%

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

   9. distribute-rgt-neg-in 51.1%

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

   10. fma-define 51.1%

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

   11. *-commutative 51.1%

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

   12. distribute-rgt-neg-in 51.1%

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

   13. metadata-eval 51.1%

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

3. Simplified 51.1%

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

5. Taylor expanded in a around 0 67.4%

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

   1. associate-*r/ 67.4%

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

   2. mul-1-neg 67.4%

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

7. Simplified 67.4%

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

8. Final simplification 67.4%

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

Reproduce

herbie shell --seed 2024148 
(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)))