Quadratic roots, medium range

Percentage Accurate: 32.1% → 95.3%

Time: 11.2s

Alternatives: 9

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: 32.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma(-2.0, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), (((-5.0 / (pow(b, 7.0) / (pow(a, 3.0) * pow(c, 4.0)))) - ((a / pow(b, 3.0)) * (c * c))) - (c / b)));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

2. Taylor expanded in a around 0 95.6%

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

4. Simplified 95.6%

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

5. Taylor expanded in c around 0 95.6%

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

   1. associate-*r/ 95.6%

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

   2. associate-/l* 95.6%

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

7. Simplified 95.6%

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

8. Final simplification 95.6%

   \[\leadsto \mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \left(\frac{-5}{\frac{{b}^{7}}{{a}^{3} \cdot {c}^{4}}} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) - \frac{c}{b}\right) \]

Alternative 2: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000:\\ \;\;\;\;\left(\sqrt{\left(b + -2 \cdot t_0\right) \cdot \left(b + 2 \cdot t_0\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -50000.0)
     (*
      (- (sqrt (* (+ b (* -2.0 t_0)) (+ b (* 2.0 t_0)))) b)
      (/ 1.0 (* a 2.0)))
     (- (/ (- c) b) (* (/ a (pow b 3.0)) (* c c))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0) {
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) * (1.0 / (a * 2.0));
	} else {
		tmp = (-c / b) - ((a / pow(b, 3.0)) * (c * c));
	}
	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((a * c))
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-50000.0d0)) then
        tmp = (sqrt(((b + ((-2.0d0) * t_0)) * (b + (2.0d0 * t_0)))) - b) * (1.0d0 / (a * 2.0d0))
    else
        tmp = (-c / b) - ((a / (b ** 3.0d0)) * (c * c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * c));
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0) {
		tmp = (Math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) * (1.0 / (a * 2.0));
	} else {
		tmp = (-c / b) - ((a / Math.pow(b, 3.0)) * (c * c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * c))
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0:
		tmp = (math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) * (1.0 / (a * 2.0))
	else:
		tmp = (-c / b) - ((a / math.pow(b, 3.0)) * (c * c))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -50000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + Float64(-2.0 * t_0)) * Float64(b + Float64(2.0 * t_0)))) - b) * Float64(1.0 / Float64(a * 2.0)));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / (b ^ 3.0)) * Float64(c * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * c));
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0)
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) * (1.0 / (a * 2.0));
	else
		tmp = (-c / b) - ((a / (b ^ 3.0)) * (c * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, 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], -50000.0], N[(N[(N[Sqrt[N[(N[(b + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $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 4 a) c)))) (*.f64 2 a)) < -5e4

   1. Initial program 84.5%

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

      1. add-sqr-sqrt 84.5%

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

      2. difference-of-squares 84.7%

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

      3. associate-*l* 84.7%

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

      4. sqrt-prod 84.7%

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

      5. metadata-eval 84.7%

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

      6. associate-*l* 84.7%

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

      7. sqrt-prod 84.7%

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

      8. metadata-eval 84.7%

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

   3. Applied egg-rr 84.7%

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

      1. *-commutative 84.7%

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

      2. cancel-sign-sub-inv 84.7%

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

      3. metadata-eval 84.7%

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

   5. Simplified 84.7%

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

      1. div-inv 84.7%

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

      2. *-commutative 84.7%

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

      3. *-commutative 84.7%

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

   7. Applied egg-rr 84.7%

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

   if -5e4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 26.5%

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

   2. Taylor expanded in b around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. unsub-neg 93.0%

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

      3. mul-1-neg 93.0%

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

      4. distribute-neg-frac 93.0%

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

      5. associate-/l* 93.0%

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

      6. associate-/r/ 93.0%

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

      7. unpow2 93.0%

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

   4. Simplified 93.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000:\\ \;\;\;\;\left(\sqrt{\left(b + -2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + 2 \cdot \sqrt{a \cdot c}\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \]

Alternative 3: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000:\\ \;\;\;\;\frac{\sqrt{\left(b + -2 \cdot t_0\right) \cdot \left(b + 2 \cdot t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -50000.0)
     (/ (- (sqrt (* (+ b (* -2.0 t_0)) (+ b (* 2.0 t_0)))) b) (* a 2.0))
     (- (/ (- c) b) (* (/ a (pow b 3.0)) (* c c))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0) {
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / pow(b, 3.0)) * (c * c));
	}
	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((a * c))
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-50000.0d0)) then
        tmp = (sqrt(((b + ((-2.0d0) * t_0)) * (b + (2.0d0 * t_0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a / (b ** 3.0d0)) * (c * c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * c));
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0) {
		tmp = (Math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / Math.pow(b, 3.0)) * (c * c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * c))
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0:
		tmp = (math.sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a / math.pow(b, 3.0)) * (c * c))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -50000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + Float64(-2.0 * t_0)) * Float64(b + Float64(2.0 * t_0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / (b ^ 3.0)) * Float64(c * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * c));
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0)
		tmp = (sqrt(((b + (-2.0 * t_0)) * (b + (2.0 * t_0)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a / (b ^ 3.0)) * (c * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, 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], -50000.0], N[(N[(N[Sqrt[N[(N[(b + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $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 4 a) c)))) (*.f64 2 a)) < -5e4

   1. Initial program 84.5%

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

      1. add-sqr-sqrt 84.5%

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

      2. difference-of-squares 84.7%

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

      3. associate-*l* 84.7%

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

      4. sqrt-prod 84.7%

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

      5. metadata-eval 84.7%

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

      6. associate-*l* 84.7%

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

      7. sqrt-prod 84.7%

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

      8. metadata-eval 84.7%

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

   3. Applied egg-rr 84.7%

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

      1. *-commutative 84.7%

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

      2. cancel-sign-sub-inv 84.7%

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

      3. metadata-eval 84.7%

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

   5. Simplified 84.7%

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

   if -5e4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 26.5%

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

   2. Taylor expanded in b around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. unsub-neg 93.0%

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

      3. mul-1-neg 93.0%

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

      4. distribute-neg-frac 93.0%

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

      5. associate-/l* 93.0%

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

      6. associate-/r/ 93.0%

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

      7. unpow2 93.0%

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

   4. Simplified 93.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000:\\ \;\;\;\;\frac{\sqrt{\left(b + -2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + 2 \cdot \sqrt{a \cdot c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \]

Alternative 4: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -50000.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a (pow b 3.0)) (* c c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / pow(b, 3.0)) * (c * c));
	}
	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)) <= -50000.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / (b ^ 3.0)) * Float64(c * c)));
	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], -50000.0], 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[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $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 4 a) c)))) (*.f64 2 a)) < -5e4

   1. Initial program 84.5%

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

   3. Simplified 84.5%

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

   if -5e4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 26.5%

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

   2. Taylor expanded in b around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. unsub-neg 93.0%

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

      3. mul-1-neg 93.0%

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

      4. distribute-neg-frac 93.0%

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

      5. associate-/l* 93.0%

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

      6. associate-/r/ 93.0%

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

      7. unpow2 93.0%

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

   4. Simplified 93.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \]

Alternative 5: 93.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (((-2.0 * (a * a)) / (pow(b, 5.0) / pow(c, 3.0))) - (c / b)) - ((a / pow(b, 3.0)) * (c * c));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((-2.0d0) * (a * a)) / ((b ** 5.0d0) / (c ** 3.0d0))) - (c / b)) - ((a / (b ** 3.0d0)) * (c * c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

2. Taylor expanded in b around inf 93.9%

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

   1. associate-+r+ 93.9%

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

   2. mul-1-neg 93.9%

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

   3. unsub-neg 93.9%

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

   4. mul-1-neg 93.9%

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

   5. unsub-neg 93.9%

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

   6. associate-/l* 93.9%

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

   7. associate-*r/ 93.9%

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

   8. unpow2 93.9%

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

   9. associate-/l* 93.9%

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

   10. associate-/r/ 93.9%

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

   11. unpow2 93.9%

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

4. Simplified 93.9%

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

5. Final simplification 93.9%

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

Alternative 6: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -50000.0)
   (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a (pow b 3.0)) (* c c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / pow(b, 3.0)) * (c * c));
	}
	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 (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-50000.0d0)) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a / (b ** 3.0d0)) * (c * c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / Math.pow(b, 3.0)) * (c * c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a / math.pow(b, 3.0)) * (c * c))
	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)) <= -50000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / (b ^ 3.0)) * Float64(c * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -50000.0)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a / (b ^ 3.0)) * (c * c));
	end
	tmp_2 = 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], -50000.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $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 4 a) c)))) (*.f64 2 a)) < -5e4

   1. Initial program 84.5%

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

   3. Simplified 84.5%

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

      1. *-commutative 84.5%

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

      2. metadata-eval 84.5%

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

      3. distribute-lft-neg-in 84.5%

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

      4. distribute-rgt-neg-in 84.5%

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

      5. *-commutative 84.5%

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

      6. fma-neg 84.5%

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

      7. associate-*l* 84.5%

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

   5. Applied egg-rr 84.5%

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

   if -5e4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 26.5%

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

   2. Taylor expanded in b around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. unsub-neg 93.0%

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

      3. mul-1-neg 93.0%

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

      4. distribute-neg-frac 93.0%

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

      5. associate-/l* 93.0%

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

      6. associate-/r/ 93.0%

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

      7. unpow2 93.0%

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

   4. Simplified 93.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -50000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \]

Alternative 7: 90.5% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

2. Taylor expanded in b around inf 19.8%

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

   1. flip-+ 19.9%

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

   2. associate-/l* 19.9%

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

   3. associate-/l* 19.9%

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

   4. associate-/l* 19.9%

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

4. Applied egg-rr 19.9%

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

   1. sqr-neg 19.9%

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

   2. associate-/r/ 19.9%

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

   3. associate-/r/ 19.9%

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

   4. associate--r+ 19.9%

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

   5. associate-/r/ 19.9%

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

6. Simplified 19.9%

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

7. Taylor expanded in b around inf 90.7%

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

   1. associate-*r* 90.7%

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

   2. *-commutative 90.7%

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

   3. associate-*l* 90.7%

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

9. Simplified 90.7%

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

10. Final simplification 90.7%

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

Alternative 8: 80.8% 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(Float64(-c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

2. Taylor expanded in b around inf 82.2%

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

   1. mul-1-neg 82.2%

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

   2. distribute-neg-frac 82.2%

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

4. Simplified 82.2%

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

5. Final simplification 82.2%

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

Alternative 9: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

   1. add-sqr-sqrt 29.9%

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

   2. difference-of-squares 29.9%

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

   3. associate-*l* 29.9%

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

   4. sqrt-prod 29.9%

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

   5. metadata-eval 29.9%

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

   6. associate-*l* 29.9%

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

   7. sqrt-prod 29.9%

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

   8. metadata-eval 29.9%

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

3. Applied egg-rr 29.9%

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

   1. *-commutative 29.9%

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

   2. cancel-sign-sub-inv 29.9%

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

   3. metadata-eval 29.9%

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

5. Simplified 29.9%

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

6. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
7. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}\right)}{a}} \]

   2. distribute-rgt-out 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{a \cdot c} \cdot \left(-2 + 2\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{a \cdot c} \cdot \color{blue}{0}\right)}{a} \]

   4. mul0-rgt 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023271 
(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)))