Quadratic roots, narrow range

Percentage Accurate: 55.3% → 91.9%

Time: 15.0s

Alternatives: 9

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% 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: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot {\left(a \cdot c\right)}^{2}\\ t_1 := {\left(a \cdot c\right)}^{4}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(\frac{t_0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot t_0\right), \left(a \cdot -4\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{4 \cdot t_1 + \mathsf{fma}\left(2, t_1 \cdot 8, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow (* a c) 2.0))) (t_1 (pow (* a c) 4.0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -2.0)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (-
      (*
       -0.25
       (+
        (+
         (/ t_0 (* a (pow b 3.0)))
         (/
          (fma 2.0 (* a (* c t_0)) (* (* a -4.0) (* c 0.0)))
          (* a (pow b 5.0))))
        (/ (+ (* 4.0 t_1) (fma 2.0 (* t_1 8.0) 0.0)) (* a (pow b 7.0)))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * pow((a * c), 2.0);
	double t_1 = pow((a * c), 4.0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -2.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-0.25 * (((t_0 / (a * pow(b, 3.0))) + (fma(2.0, (a * (c * t_0)), ((a * -4.0) * (c * 0.0))) / (a * pow(b, 5.0)))) + (((4.0 * t_1) + fma(2.0, (t_1 * 8.0), 0.0)) / (a * pow(b, 7.0))))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * (Float64(a * c) ^ 2.0))
	t_1 = Float64(a * c) ^ 4.0
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -2.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-0.25 * Float64(Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(fma(2.0, Float64(a * Float64(c * t_0)), Float64(Float64(a * -4.0) * Float64(c * 0.0))) / Float64(a * (b ^ 5.0)))) + Float64(Float64(Float64(4.0 * t_1) + fma(2.0, Float64(t_1 * 8.0), 0.0)) / Float64(a * (b ^ 7.0))))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.0], 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[(-0.25 * N[(N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(a * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -4.0), $MachinePrecision] * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 * t$95$1), $MachinePrecision] + N[(2.0 * N[(t$95$1 * 8.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $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)) < -2

   1. Initial program 85.9%

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

   3. Simplified 86.1%

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

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

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

      1. flip3-- 51.5%

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

      2. div-inv 51.5%

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

      3. pow2 51.5%

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

      4. pow-pow 51.4%

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

      5. metadata-eval 51.4%

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

      6. associate-*l* 51.4%

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

      7. unpow-prod-down 51.4%

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

      8. metadata-eval 51.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      9. pow2 51.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      10. pow2 51.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      11. pow-prod-up 51.5%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      12. metadata-eval 51.5%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      13. distribute-rgt-out 51.5%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}}}{a \cdot 2} \]

   5. Applied egg-rr 51.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in b around inf 94.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(2 \cdot \left(a \cdot \left(c \cdot \left(-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.25 \cdot \frac{-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.25 \cdot \frac{-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

   8. Simplified 94.5%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, \left(a \cdot c\right) \cdot \mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b}} \]

   9. Taylor expanded in a around 0 94.5%

      \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, \color{blue}{8 \cdot \left({a}^{4} \cdot {c}^{4}\right)}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]
   10. Step-by-step derivation

       1. metadata-eval 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left({a}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {c}^{4}\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       2. pow-sqr 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left(\color{blue}{\left({a}^{2} \cdot {a}^{2}\right)} \cdot {c}^{4}\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       3. metadata-eval 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {c}^{\color{blue}{\left(2 \cdot 2\right)}}\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       4. pow-sqr 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left(\left({a}^{2} \cdot {a}^{2}\right) \cdot \color{blue}{\left({c}^{2} \cdot {c}^{2}\right)}\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       5. swap-sqr 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       6. unpow2 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       7. unpow2 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       8. swap-sqr 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left(\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       9. unpow2 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       10. unpow2 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right)\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       11. unpow2 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       12. swap-sqr 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       13. unpow2 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}\right), 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       14. pow-sqr 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       15. metadata-eval 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{\color{blue}{4}}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

   11. Simplified 94.5%

       \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(2, \color{blue}{8 \cdot {\left(a \cdot c\right)}^{4}}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

   12. Taylor expanded in a around 0 94.5%

       \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(\color{blue}{4 \cdot \left({a}^{4} \cdot {c}^{4}\right)} + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]
   13. Step-by-step derivation

       1. metadata-eval 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left({a}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {c}^{4}\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       2. pow-sqr 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left(\color{blue}{\left({a}^{2} \cdot {a}^{2}\right)} \cdot {c}^{4}\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       3. metadata-eval 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {c}^{\color{blue}{\left(2 \cdot 2\right)}}\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       4. pow-sqr 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left(\left({a}^{2} \cdot {a}^{2}\right) \cdot \color{blue}{\left({c}^{2} \cdot {c}^{2}\right)}\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       5. swap-sqr 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)} + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       6. unpow2 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       7. unpow2 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       8. swap-sqr 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left(\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       9. unpow2 94.5%

          \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       10. unpow2 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right)\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       11. unpow2 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       12. swap-sqr 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       13. unpow2 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}\right) + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       14. pow-sqr 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

       15. metadata-eval 94.5%

           \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(4 \cdot {\left(a \cdot c\right)}^{\color{blue}{4}} + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]

   14. Simplified 94.5%

       \[\leadsto -0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left(\color{blue}{4 \cdot {\left(a \cdot c\right)}^{4}} + 0\right) + \mathsf{fma}\left(2, 8 \cdot {\left(a \cdot c\right)}^{4}, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right), \left(a \cdot -4\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{4 \cdot {\left(a \cdot c\right)}^{4} + \mathsf{fma}\left(2, {\left(a \cdot c\right)}^{4} \cdot 8, 0\right)}{a \cdot {b}^{7}}\right) - \frac{c}{b}\\ \end{array} \]

Alternative 2: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -2.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (+
    (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (-
     (-
      (* -0.25 (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0))))
      (/ (* a (pow c 2.0)) (pow b 3.0)))
     (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -2.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -2.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.0], 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[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 85.9%

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

   3. Simplified 86.1%

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

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

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in b around inf 94.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

   5. Taylor expanded in c around 0 94.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 94.4%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left(4 \cdot {a}^{4}\right) \cdot {c}^{4} + \left(16 \cdot {a}^{4}\right) \cdot {c}^{4}}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 94.4%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{4 \cdot \left({a}^{4} \cdot {c}^{4}\right)} + \left(16 \cdot {a}^{4}\right) \cdot {c}^{4}}{a \cdot {b}^{7}}\right)\right) \]

      3. associate-*r* 94.4%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}}{a \cdot {b}^{7}}\right)\right) \]

      4. distribute-rgt-out 94.4%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}\right)\right) \]

      5. times-frac 94.4%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}\right)}\right)\right) \]

   7. Simplified 94.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 3: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot {\left(a \cdot c\right)}^{2}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{t_0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot t_0\right), \left(a \cdot -4\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow (* a c) 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -2.0)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
     (fma
      -0.25
      (+
       (/ t_0 (* a (pow b 3.0)))
       (/
        (fma 2.0 (* a (* c t_0)) (* (* a -4.0) (* c 0.0)))
        (* a (pow b 5.0))))
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * pow((a * c), 2.0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -2.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(-0.25, ((t_0 / (a * pow(b, 3.0))) + (fma(2.0, (a * (c * t_0)), ((a * -4.0) * (c * 0.0))) / (a * pow(b, 5.0)))), (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * (Float64(a * c) ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -2.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = fma(-0.25, Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(fma(2.0, Float64(a * Float64(c * t_0)), Float64(Float64(a * -4.0) * Float64(c * 0.0))) / Float64(a * (b ^ 5.0)))), Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.0], 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[(-0.25 * N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(a * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -4.0), $MachinePrecision] * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $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)) < -2

   1. Initial program 85.9%

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

   3. Simplified 86.1%

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

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

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

      1. flip3-- 51.5%

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

      2. div-inv 51.5%

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

      3. pow2 51.5%

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

      4. pow-pow 51.4%

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

      5. metadata-eval 51.4%

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

      6. associate-*l* 51.4%

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

      7. unpow-prod-down 51.4%

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

      8. metadata-eval 51.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      9. pow2 51.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      10. pow2 51.4%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      11. pow-prod-up 51.5%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      12. metadata-eval 51.5%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      13. distribute-rgt-out 51.5%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}}}{a \cdot 2} \]

   5. Applied egg-rr 51.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in b around inf 91.6%

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

   8. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), \left(-4 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}, \frac{-c}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{4 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right), \left(a \cdot -4\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}, \frac{-c}{b}\right)\\ \end{array} \]

Alternative 4: 89.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -2.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (- (/ (* -2.0 (* (pow a 2.0) (pow c 3.0))) (pow b 5.0)) (/ c b))
    (/ a (/ (pow b 3.0) (pow c 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -2.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (pow(a, 2.0) * pow(c, 3.0))) / pow(b, 5.0)) - (c / b)) - (a / (pow(b, 3.0) / pow(c, 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -2.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((a ^ 2.0) * (c ^ 3.0))) / (b ^ 5.0)) - Float64(c / b)) - Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.0], 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[(N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[Power[c, 2.0], $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 4 a) c)))) (*.f64 2 a)) < -2

   1. Initial program 85.9%

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

   3. Simplified 86.1%

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

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

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in b around inf 91.6%

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

      1. associate-+r+ 91.6%

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

      2. mul-1-neg 91.6%

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

      3. unsub-neg 91.6%

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

      4. mul-1-neg 91.6%

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

      5. unsub-neg 91.6%

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

      6. associate-*r/ 91.6%

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

      7. *-commutative 91.6%

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

      8. associate-/l* 91.6%

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

   6. Simplified 91.6%

      \[\leadsto \color{blue}{\left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

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

Alternative 5: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.001752:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(a \cdot {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)}{b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.001752)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (* -0.25 (* a (pow (* -2.0 (/ c b)) 2.0))) b) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.001752) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((-0.25 * (a * pow((-2.0 * (c / b)), 2.0))) / b) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.001752)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-0.25 * Float64(a * (Float64(-2.0 * Float64(c / b)) ^ 2.0))) / b) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.001752], 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[(N[(-0.25 * N[(a * N[Power[N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(c / b), $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)) < -0.0017520000000000001

   1. Initial program 79.4%

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

   3. Simplified 79.6%

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

   if -0.0017520000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.1%

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

      1. *-commutative 46.1%

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

   3. Simplified 46.1%

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

      1. flip3-- 45.9%

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

      2. div-inv 46.0%

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

      3. pow2 46.0%

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

      4. pow-pow 45.8%

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

      5. metadata-eval 45.8%

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

      6. associate-*l* 45.8%

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

      7. unpow-prod-down 45.8%

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

      8. metadata-eval 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      9. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      10. pow2 45.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      11. pow-prod-up 46.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      12. metadata-eval 46.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      13. distribute-rgt-out 46.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}}}{a \cdot 2} \]

   5. Applied egg-rr 46.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in a around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

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

   8. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \left({\left(-2 \cdot \frac{c}{b}\right)}^{2} + 0\right)\right) \cdot -0.25}{b} - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.001752:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(a \cdot {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)}{b} - \frac{c}{b}\\ \end{array} \]

Alternative 6: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.001752:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(a \cdot {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)}{b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.001752)
     t_0
     (- (/ (* -0.25 (* a (pow (* -2.0 (/ c b)) 2.0))) b) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.001752) {
		tmp = t_0;
	} else {
		tmp = ((-0.25 * (a * pow((-2.0 * (c / b)), 2.0))) / b) - (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) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.001752d0)) then
        tmp = t_0
    else
        tmp = (((-0.25d0) * (a * (((-2.0d0) * (c / b)) ** 2.0d0))) / b) - (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) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.001752) {
		tmp = t_0;
	} else {
		tmp = ((-0.25 * (a * Math.pow((-2.0 * (c / b)), 2.0))) / b) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.001752:
		tmp = t_0
	else:
		tmp = ((-0.25 * (a * math.pow((-2.0 * (c / b)), 2.0))) / b) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.001752)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-0.25 * Float64(a * (Float64(-2.0 * Float64(c / b)) ^ 2.0))) / b) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.001752)
		tmp = t_0;
	else
		tmp = ((-0.25 * (a * ((-2.0 * (c / b)) ^ 2.0))) / b) - (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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.001752], t$95$0, N[(N[(N[(-0.25 * N[(a * N[Power[N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(c / b), $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)) < -0.0017520000000000001

   1. Initial program 79.4%

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

   if -0.0017520000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 46.1%

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

      1. *-commutative 46.1%

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

   3. Simplified 46.1%

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

      1. flip3-- 45.9%

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

      2. div-inv 46.0%

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

      3. pow2 46.0%

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

      4. pow-pow 45.8%

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

      5. metadata-eval 45.8%

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

      6. associate-*l* 45.8%

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

      7. unpow-prod-down 45.8%

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

      8. metadata-eval 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      9. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      10. pow2 45.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      11. pow-prod-up 46.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      12. metadata-eval 46.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

      13. distribute-rgt-out 46.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}}}{a \cdot 2} \]

   5. Applied egg-rr 46.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

   6. Taylor expanded in a around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. mul-1-neg 89.0%

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

      3. unsub-neg 89.0%

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

   8. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{\left(a \cdot \left({\left(-2 \cdot \frac{c}{b}\right)}^{2} + 0\right)\right) \cdot -0.25}{b} - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.001752:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(a \cdot {\left(-2 \cdot \frac{c}{b}\right)}^{2}\right)}{b} - \frac{c}{b}\\ \end{array} \]

Alternative 7: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -4.5e-7) t_0 (/ (- c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -4.5e-7) {
		tmp = t_0;
	} else {
		tmp = -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) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-4.5d-7)) then
        tmp = t_0
    else
        tmp = -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) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -4.5e-7) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -4.5e-7:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -4.5e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -4.5e-7)
		tmp = t_0;
	else
		tmp = -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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4.5e-7], t$95$0, N[((-c) / b), $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)) < -4.4999999999999998e-7

   1. Initial program 73.7%

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

   if -4.4999999999999998e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 29.3%

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

      1. *-commutative 29.3%

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

   3. Simplified 29.3%

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

   4. Taylor expanded in b around inf 85.5%

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

      1. mul-1-neg 85.5%

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

      2. distribute-neg-frac 85.5%

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

   6. Simplified 85.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 64.5% 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 57.7%

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

   1. *-commutative 57.7%

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

3. Simplified 57.7%

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

4. Taylor expanded in b around inf 62.1%

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

   1. mul-1-neg 62.1%

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

   2. distribute-neg-frac 62.1%

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

6. Simplified 62.1%

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

7. Final simplification 62.1%

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

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 57.7%

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

   1. *-commutative 57.7%

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

3. Simplified 57.7%

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

   1. flip3-- 57.5%

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

   2. div-inv 57.5%

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

   3. pow2 57.5%

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

   4. pow-pow 57.2%

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

   5. metadata-eval 57.2%

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

   6. associate-*l* 57.2%

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

   7. unpow-prod-down 57.2%

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

   8. metadata-eval 57.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   9. pow2 57.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   10. pow2 57.2%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   11. pow-prod-up 57.4%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   12. metadata-eval 57.4%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   13. distribute-rgt-out 57.4%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}}}{a \cdot 2} \]

5. Applied egg-rr 57.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. add-cube-cbrt 57.4%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 2}} \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 2}}\right) \cdot \sqrt[3]{\frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 2}}} \]

   2. pow3 57.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) + \sqrt{\left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 2}}\right)}^{3}} \]

7. Applied egg-rr 57.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(a \cdot c\right), \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right), {b}^{4}\right)}}\right)}{a \cdot 2}}\right)}^{3}} \]

8. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \left({1}^{0.3333333333333333} \cdot \frac{b + -1 \cdot b}{a}\right)} \]
9. Step-by-step derivation

   1. pow-base-1 3.2%

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

   2. *-lft-identity 3.2%

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

   3. distribute-rgt1-in 3.2%

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

   4. metadata-eval 3.2%

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

   5. mul0-lft 3.2%

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

   6. associate-*r/ 3.2%

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

   7. metadata-eval 3.2%

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

10. Simplified 3.2%

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

11. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023308 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))