Quadratic roots, medium range

Percentage Accurate: 31.7% → 95.4%

Time: 16.2s

Alternatives: 10

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
      (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
    (/ (* c c) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - ((c * c) / pow(b, 3.0)))) - (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 = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))))) - ((c * c) / (b ** 3.0d0)))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))) - ((c * c) / Math.pow(b, 3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0))))) - ((c * c) / math.pow(b, 3.0)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64(Float64(c * c) / (b ^ 3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-5.0 * ((a * (c ^ 4.0)) / (b ^ 7.0))))) - ((c * c) / (b ^ 3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

5. Taylor expanded in a around 0 97.0%

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

6. Taylor expanded in c around 0 97.0%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
7. Step-by-step derivation

   1. unpow2 97.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

8. Applied egg-rr 97.0%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{\color{blue}{c \cdot c}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
9. Step-by-step derivation

   1. associate-*r/ 97.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} + a \cdot \left(-1 \cdot \frac{c \cdot c}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   2. neg-mul-1 97.0%

      \[\leadsto \frac{\color{blue}{-c}}{b} + a \cdot \left(-1 \cdot \frac{c \cdot c}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

10. Applied egg-rr 97.0%

    \[\leadsto \color{blue}{\frac{-c}{b}} + a \cdot \left(-1 \cdot \frac{c \cdot c}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

11. Final simplification 97.0%

    \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b} \]
12. Add Preprocessing

Alternative 2: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -10000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -10000.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -10000.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (c / -b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -10000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -10000.0], 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[(c / (-b)), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1e4

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

   3. Simplified 84.4%

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

   if -1e4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 24.8%

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

      1. *-commutative 24.8%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in a around 0 95.0%

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

      1. mul-1-neg 95.0%

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

      2. unsub-neg 95.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 95.0%

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

      4. distribute-neg-frac2 95.0%

         \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 95.0%

         \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   7. Simplified 95.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -10000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -10000.0)
     t_0
     (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-10000.0d0)) then
        tmp = t_0
    else
        tmp = (c / -b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * (Math.pow(c, 2.0) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -10000.0:
		tmp = t_0
	else:
		tmp = (c / -b) - (a * (math.pow(c, 2.0) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -10000.0)
		tmp = t_0;
	else
		tmp = (c / -b) - (a * ((c ^ 2.0) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], t$95$0, N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1e4

   1. Initial program 83.9%

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

   if -1e4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 24.8%

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

      1. *-commutative 24.8%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in a around 0 95.0%

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

      1. mul-1-neg 95.0%

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

      2. unsub-neg 95.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 95.0%

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

      4. distribute-neg-frac2 95.0%

         \[\leadsto \color{blue}{\frac{c}{-b}} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. associate-/l* 95.0%

         \[\leadsto \frac{c}{-b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

   7. Simplified 95.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -10000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - {\left(\frac{c}{-b}\right)}^{2}\right)}{b} - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (/
   (* a (- (* -2.0 (/ (* a (pow c 3.0)) (pow b 4.0))) (pow (/ c (- b)) 2.0)))
   b)
  (/ c b)))

C

double code(double a, double b, double c) {
	return ((a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 4.0))) - pow((c / -b), 2.0))) / b) - (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 = ((a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 4.0d0))) - ((c / -b) ** 2.0d0))) / b) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return ((a * ((-2.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) - Math.pow((c / -b), 2.0))) / b) - (c / b);
}

Python

def code(a, b, c):
	return ((a * ((-2.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 4.0))) - math.pow((c / -b), 2.0))) / b) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0))) - (Float64(c / Float64(-b)) ^ 2.0))) / b) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a * ((-2.0 * ((a * (c ^ 3.0)) / (b ^ 4.0))) - ((c / -b) ^ 2.0))) / b) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

5. Taylor expanded in b around inf 95.3%

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

6. Taylor expanded in a around 0 95.3%

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

   1. neg-mul-1 95.3%

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

   2. +-commutative 95.3%

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

   3. unsub-neg 95.3%

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

8. Simplified 95.3%

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

   1. div-sub 95.4%

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

   2. associate-/l* 95.4%

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

   3. *-commutative 95.4%

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

10. Applied egg-rr 95.4%

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

11. Final simplification 95.4%

    \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - {\left(\frac{c}{-b}\right)}^{2}\right)}{b} - \frac{c}{b} \]
12. Add Preprocessing

Alternative 5: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -10000.0) t_0 (/ (- (* a (- (pow (/ c b) 2.0))) c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = t_0;
	} else {
		tmp = ((a * -pow((c / b), 2.0)) - c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-10000.0d0)) then
        tmp = t_0
    else
        tmp = ((a * -((c / b) ** 2.0d0)) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -10000.0) {
		tmp = t_0;
	} else {
		tmp = ((a * -Math.pow((c / b), 2.0)) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -10000.0:
		tmp = t_0
	else:
		tmp = ((a * -math.pow((c / b), 2.0)) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -10000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a * Float64(-(Float64(c / b) ^ 2.0))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -10000.0)
		tmp = t_0;
	else
		tmp = ((a * -((c / b) ^ 2.0)) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], t$95$0, N[(N[(N[(a * (-N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1e4

   1. Initial program 83.9%

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

   if -1e4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 24.8%

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

      1. *-commutative 24.8%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in b around inf 97.1%

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

   6. Taylor expanded in a around 0 97.1%

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

      1. neg-mul-1 97.1%

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

      2. +-commutative 97.1%

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

      3. unsub-neg 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in a around 0 95.0%

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

   11. Simplified 95.0%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -10000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{4}} - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (* a (- (/ (* -2.0 (* a (pow c 3.0))) (pow b 4.0)) (* (/ c b) (/ c b))))
   c)
  b))

C

double code(double a, double b, double c) {
	return ((a * (((-2.0 * (a * pow(c, 3.0))) / pow(b, 4.0)) - ((c / b) * (c / b)))) - 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 = ((a * ((((-2.0d0) * (a * (c ** 3.0d0))) / (b ** 4.0d0)) - ((c / b) * (c / b)))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((a * (((-2.0 * (a * Math.pow(c, 3.0))) / Math.pow(b, 4.0)) - ((c / b) * (c / b)))) - c) / b;
}

Python

def code(a, b, c):
	return ((a * (((-2.0 * (a * math.pow(c, 3.0))) / math.pow(b, 4.0)) - ((c / b) * (c / b)))) - c) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(Float64(Float64(-2.0 * Float64(a * (c ^ 3.0))) / (b ^ 4.0)) - Float64(Float64(c / b) * Float64(c / b)))) - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a * (((-2.0 * (a * (c ^ 3.0))) / (b ^ 4.0)) - ((c / b) * (c / b)))) - c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(N[(N[(-2.0 * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

5. Taylor expanded in b around inf 95.3%

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

6. Taylor expanded in a around 0 95.3%

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

   1. neg-mul-1 95.3%

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

   2. +-commutative 95.3%

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

   3. unsub-neg 95.3%

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

8. Simplified 95.3%

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

   1. unpow2 95.3%

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

10. Applied egg-rr 95.3%

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

11. Final simplification 95.3%

    \[\leadsto \frac{a \cdot \left(\frac{-2 \cdot \left(a \cdot {c}^{3}\right)}{{b}^{4}} - \frac{c}{b} \cdot \frac{c}{b}\right) - c}{b} \]
12. Add Preprocessing

Alternative 7: 93.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-1.0d0) / (a * (b * (((1.0d0 / a) / c) - ((1.0d0 / (b ** 2.0d0)) + ((c * a) / (b ** 4.0d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

   1. add-cbrt-cube 28.6%

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

   2. pow1/3 29.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 29.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 29.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 29.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 28.7%

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

8. Simplified 28.7%

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

   1. clear-num 28.7%

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

   2. inv-pow 28.7%

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

   3. neg-mul-1 28.7%

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

   4. fma-define 28.7%

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

   5. pow1/3 29.7%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   6. pow-pow 28.8%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   7. metadata-eval 28.8%

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

   8. associate-*l* 28.8%

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

10. Applied egg-rr 28.8%

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

    1. unpow-1 28.8%

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

    2. associate-/l* 28.8%

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

    3. sub-neg 28.8%

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

    4. +-commutative 28.8%

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

    5. distribute-lft-neg-in 28.8%

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

    6. metadata-eval 28.8%

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

    7. fma-define 28.8%

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

12. Simplified 28.8%

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

13. Taylor expanded in b around inf 95.0%

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

    1. +-commutative 95.0%

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

    2. mul-1-neg 95.0%

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

    3. unsub-neg 95.0%

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

    4. distribute-lft1-in 95.0%

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

    5. metadata-eval 95.0%

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

    6. mul-1-neg 95.0%

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

    7. *-commutative 95.0%

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

    8. associate-/r* 95.0%

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

15. Simplified 95.0%

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

16. Final simplification 95.0%

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

Alternative 8: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

5. Taylor expanded in b around inf 95.3%

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

6. Taylor expanded in a around 0 95.3%

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

   1. neg-mul-1 95.3%

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

   2. +-commutative 95.3%

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

   3. unsub-neg 95.3%

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

8. Simplified 95.3%

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

9. Taylor expanded in a around 0 92.4%

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

11. Simplified 92.4%

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

12. Final simplification 92.4%

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

Alternative 9: 81.1% 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 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

5. Taylor expanded in b around inf 83.5%

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

   1. associate-*r/ 83.5%

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

   2. mul-1-neg 83.5%

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

7. Simplified 83.5%

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

8. Final simplification 83.5%

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

Alternative 10: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

   1. add-cbrt-cube 28.6%

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

   2. pow1/3 29.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 29.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 29.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 29.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 29.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 28.7%

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

8. Simplified 28.7%

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

   1. clear-num 28.7%

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

   2. inv-pow 28.7%

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

   3. neg-mul-1 28.7%

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

   4. fma-define 28.7%

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

   5. pow1/3 29.7%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   6. pow-pow 28.8%

      \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}\right)}^{-1} \]

   7. metadata-eval 28.8%

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

   8. associate-*l* 28.8%

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

10. Applied egg-rr 28.8%

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

    1. unpow-1 28.8%

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

    2. associate-/l* 28.8%

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

    3. sub-neg 28.8%

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

    4. +-commutative 28.8%

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

    5. distribute-lft-neg-in 28.8%

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

    6. metadata-eval 28.8%

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

    7. fma-define 28.8%

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

12. Simplified 28.8%

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

13. Taylor expanded in a around 0 3.2%

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

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

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

    4. mul0-lft 3.2%

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

    5. metadata-eval 3.2%

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

15. Simplified 3.2%

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

16. Taylor expanded in a around 0 3.2%

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

Reproduce

herbie shell --seed 2024157 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))