Quadratic roots, narrow range

Percentage Accurate: 56.0% → 91.5%

Time: 16.0s

Alternatives: 11

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 56.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.3242:\\ \;\;\;\;\frac{\frac{{b}^{2} - t_0}{\frac{a}{-0.5}}}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 4 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} - \frac{\frac{c}{b}}{\frac{b}{c}} \cdot \frac{a}{b}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.3242)
     (/ (/ (- (pow b 2.0) t_0) (/ a -0.5)) (+ b (sqrt t_0)))
     (+
      (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (-
       (-
        (*
         -0.25
         (/
          (+ (* 16.0 (* (pow a 4.0) (pow c 4.0))) (* 4.0 (pow (* a c) 4.0)))
          (* a (pow b 7.0))))
        (* (/ (/ c b) (/ b c)) (/ a b)))
       (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.3242) {
		tmp = ((pow(b, 2.0) - t_0) / (a / -0.5)) / (b + sqrt(t_0));
	} else {
		tmp = (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * (((16.0 * (pow(a, 4.0) * pow(c, 4.0))) + (4.0 * pow((a * c), 4.0))) / (a * pow(b, 7.0)))) - (((c / b) / (b / c)) * (a / b))) - (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.3242)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_0) / Float64(a / -0.5)) / Float64(b + sqrt(t_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(16.0 * Float64((a ^ 4.0) * (c ^ 4.0))) + Float64(4.0 * (Float64(a * c) ^ 4.0))) / Float64(a * (b ^ 7.0)))) - Float64(Float64(Float64(c / b) / Float64(b / c)) * Float64(a / b))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[Power[b, 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], -0.3242], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $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[(16.0 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c / b), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] * N[(a / b), $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)) < -0.324199999999999988

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

   3. Simplified 85.2%

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

      1. pow1/2 85.2%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}{a \cdot 2} \]

      2. pow-to-exp 80.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{a \cdot 2} \]

      3. fma-neg 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)} \cdot 0.5}}{a \cdot 2} \]

      4. distribute-lft-neg-in 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      5. *-commutative 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      6. distribute-lft-neg-in 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      7. metadata-eval 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)\right) \cdot 0.5}}{a \cdot 2} \]

      8. associate-*r* 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right) \cdot a}\right)\right) \cdot 0.5}}{a \cdot 2} \]

   5. Applied egg-rr 81.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. frac-2neg 81.1%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)}{-a \cdot 2}} \]

      2. div-inv 81.0%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   7. Applied egg-rr 85.3%

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

      1. flip-- 84.7%

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

      2. frac-2neg 84.7%

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

      3. distribute-rgt-neg-in 84.7%

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

      4. metadata-eval 84.7%

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

      5. frac-times 84.6%

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

   9. Applied egg-rr 85.0%

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

       1. *-commutative 85.0%

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

       2. associate-/l/ 85.1%

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

       3. *-commutative 85.1%

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

       4. associate-/l* 85.1%

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

       5. fma-udef 86.2%

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

       6. unpow2 86.2%

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

       7. +-commutative 86.2%

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

       8. fma-def 86.3%

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

       9. *-commutative 86.3%

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

       10. associate-/l* 86.3%

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

       11. metadata-eval 86.3%

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

       12. fma-udef 86.2%

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

   11. Simplified 86.2%

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

   if -0.324199999999999988 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 48.5%

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

      1. *-commutative 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in b around inf 96.3%

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

      1. unpow2 96.3%

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

      2. *-commutative 96.3%

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

      3. *-commutative 96.3%

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

      4. swap-sqr 96.3%

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

      5. pow-prod-down 96.3%

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

      6. pow-prod-down 96.3%

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

      7. pow-sqr 96.3%

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

      8. metadata-eval 96.3%

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

      9. metadata-eval 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. *-commutative 96.3%

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

      2. unpow3 96.3%

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

      3. times-frac 96.3%

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

      4. unpow2 96.3%

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

      5. frac-times 96.3%

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

      6. pow1 96.3%

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

      7. metadata-eval 96.3%

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

      8. pow1 96.3%

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

      9. metadata-eval 96.3%

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

      10. pow-sqr 96.3%

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

      11. metadata-eval 96.3%

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

      12. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. unpow2 96.3%

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

      2. clear-num 96.3%

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

      3. un-div-inv 96.3%

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

   10. Applied egg-rr 96.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.3242:\\ \;\;\;\;\frac{\frac{{b}^{2} - \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}{\frac{a}{-0.5}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 4 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} - \frac{\frac{c}{b}}{\frac{b}{c}} \cdot \frac{a}{b}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 2: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.3242:\\ \;\;\;\;\frac{\frac{{b}^{2} - t_0}{\frac{a}{-0.5}}}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot {c}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.3242)
     (/ (/ (- (pow b 2.0) t_0) (/ a -0.5)) (+ b (sqrt t_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 t_0 = fma(c, (a * -4.0), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.3242) {
		tmp = ((pow(b, 2.0) - t_0) / (a / -0.5)) / (b + sqrt(t_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)
	t_0 = fma(c, Float64(a * -4.0), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.3242)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_0) / Float64(a / -0.5)) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((a ^ 2.0) * (c ^ 3.0))) / (b ^ 5.0)) - Float64(c / b)) - Float64(Float64(a / (b ^ 3.0)) * (c ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[Power[b, 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], -0.3242], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(a / -0.5), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $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[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 2.0], $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)) < -0.324199999999999988

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

   3. Simplified 85.2%

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

      1. pow1/2 85.2%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}{a \cdot 2} \]

      2. pow-to-exp 80.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{a \cdot 2} \]

      3. fma-neg 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)} \cdot 0.5}}{a \cdot 2} \]

      4. distribute-lft-neg-in 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      5. *-commutative 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      6. distribute-lft-neg-in 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      7. metadata-eval 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)\right) \cdot 0.5}}{a \cdot 2} \]

      8. associate-*r* 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right) \cdot a}\right)\right) \cdot 0.5}}{a \cdot 2} \]

   5. Applied egg-rr 81.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. frac-2neg 81.1%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)}{-a \cdot 2}} \]

      2. div-inv 81.0%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   7. Applied egg-rr 85.3%

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

      1. flip-- 84.7%

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

      2. frac-2neg 84.7%

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

      3. distribute-rgt-neg-in 84.7%

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

      4. metadata-eval 84.7%

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

      5. frac-times 84.6%

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

   9. Applied egg-rr 85.0%

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

       1. *-commutative 85.0%

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

       2. associate-/l/ 85.1%

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

       3. *-commutative 85.1%

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

       4. associate-/l* 85.1%

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

       5. fma-udef 86.2%

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

       6. unpow2 86.2%

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

       7. +-commutative 86.2%

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

       8. fma-def 86.3%

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

       9. *-commutative 86.3%

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

       10. associate-/l* 86.3%

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

       11. metadata-eval 86.3%

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

       12. fma-udef 86.2%

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

   11. Simplified 86.2%

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

   if -0.324199999999999988 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 48.5%

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

      1. *-commutative 48.5%

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

   3. Simplified 48.5%

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

      1. pow1/2 48.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}{a \cdot 2} \]

      2. pow-to-exp 45.6%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{a \cdot 2} \]

      3. fma-neg 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)} \cdot 0.5}}{a \cdot 2} \]

      4. distribute-lft-neg-in 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      5. *-commutative 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      6. distribute-lft-neg-in 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      7. metadata-eval 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)\right) \cdot 0.5}}{a \cdot 2} \]

      8. associate-*r* 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right) \cdot a}\right)\right) \cdot 0.5}}{a \cdot 2} \]

   5. Applied egg-rr 45.6%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. frac-2neg 45.6%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)}{-a \cdot 2}} \]

      2. div-inv 45.6%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   7. Applied egg-rr 48.7%

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

   8. Taylor expanded in b around inf 94.0%

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

      1. associate-+r+ 94.0%

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

      2. mul-1-neg 94.0%

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

      3. unsub-neg 94.0%

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

      4. mul-1-neg 94.0%

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

      5. unsub-neg 94.0%

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

      6. associate-*r/ 94.0%

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

      7. *-commutative 94.0%

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

      8. associate-/l* 94.0%

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

      9. associate-/r/ 94.0%

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

   10. Simplified 94.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.3242:\\ \;\;\;\;\frac{\frac{{b}^{2} - \mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}{\frac{a}{-0.5}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, {b}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot {c}^{2}\\ \end{array} \]

Alternative 3: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.3242:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{1}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot {c}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.3242)
   (* (- b (sqrt (fma b b (* a (* c -4.0))))) (/ 1.0 (* 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)) <= -0.3242) {
		tmp = (b - sqrt(fma(b, b, (a * (c * -4.0))))) * (1.0 / (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)) <= -0.3242)
		tmp = Float64(Float64(b - sqrt(fma(b, b, Float64(a * Float64(c * -4.0))))) * Float64(1.0 / 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(Float64(a / (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], -0.3242], N[(N[(b - N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * -2.0), $MachinePrecision]), $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[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 2.0], $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)) < -0.324199999999999988

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

   3. Simplified 85.2%

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

      1. pow1/2 85.2%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}{a \cdot 2} \]

      2. pow-to-exp 80.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{a \cdot 2} \]

      3. fma-neg 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)} \cdot 0.5}}{a \cdot 2} \]

      4. distribute-lft-neg-in 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      5. *-commutative 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      6. distribute-lft-neg-in 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      7. metadata-eval 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)\right) \cdot 0.5}}{a \cdot 2} \]

      8. associate-*r* 81.1%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right) \cdot a}\right)\right) \cdot 0.5}}{a \cdot 2} \]

   5. Applied egg-rr 81.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. frac-2neg 81.1%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)}{-a \cdot 2}} \]

      2. div-inv 81.0%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   7. Applied egg-rr 85.3%

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

   if -0.324199999999999988 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 48.5%

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

      1. *-commutative 48.5%

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

   3. Simplified 48.5%

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

      1. pow1/2 48.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}{a \cdot 2} \]

      2. pow-to-exp 45.6%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{a \cdot 2} \]

      3. fma-neg 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)} \cdot 0.5}}{a \cdot 2} \]

      4. distribute-lft-neg-in 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      5. *-commutative 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      6. distribute-lft-neg-in 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      7. metadata-eval 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)\right) \cdot 0.5}}{a \cdot 2} \]

      8. associate-*r* 45.6%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right) \cdot a}\right)\right) \cdot 0.5}}{a \cdot 2} \]

   5. Applied egg-rr 45.6%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. frac-2neg 45.6%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)}{-a \cdot 2}} \]

      2. div-inv 45.6%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   7. Applied egg-rr 48.7%

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

   8. Taylor expanded in b around inf 94.0%

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

      1. associate-+r+ 94.0%

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

      2. mul-1-neg 94.0%

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

      3. unsub-neg 94.0%

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

      4. mul-1-neg 94.0%

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

      5. unsub-neg 94.0%

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

      6. associate-*r/ 94.0%

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

      7. *-commutative 94.0%

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

      8. associate-/l* 94.0%

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

      9. associate-/r/ 94.0%

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

   10. Simplified 94.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.3242:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right) \cdot \frac{1}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot {c}^{2}\\ \end{array} \]

Alternative 4: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{{b}^{2} - t_0}{b + \sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.01)
     (* (/ 1.0 (* a -2.0)) (/ (- (pow b 2.0) t_0) (+ b (sqrt t_0))))
     (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.01) {
		tmp = (1.0 / (a * -2.0)) * ((pow(b, 2.0) - t_0) / (b + sqrt(t_0)));
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.01)
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(Float64((b ^ 2.0) - t_0) / Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $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], -0.01], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $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)) < -0.0100000000000000002

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

   3. Simplified 82.1%

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

      1. pow1/2 82.1%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}{a \cdot 2} \]

      2. pow-to-exp 77.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{a \cdot 2} \]

      3. fma-neg 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)} \cdot 0.5}}{a \cdot 2} \]

      4. distribute-lft-neg-in 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      5. *-commutative 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      6. distribute-lft-neg-in 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      7. metadata-eval 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)\right) \cdot 0.5}}{a \cdot 2} \]

      8. associate-*r* 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right) \cdot a}\right)\right) \cdot 0.5}}{a \cdot 2} \]

   5. Applied egg-rr 78.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. frac-2neg 78.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)}{-a \cdot 2}} \]

      2. div-inv 77.9%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   7. Applied egg-rr 82.2%

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

      1. flip-- 82.0%

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

      2. pow1 82.0%

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

      3. pow1 82.0%

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

      4. pow-sqr 82.0%

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

      5. metadata-eval 82.0%

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

      6. add-sqr-sqrt 82.6%

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

      7. associate-*r* 82.6%

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

      8. *-commutative 82.6%

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

      9. associate-*l* 82.6%

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

      10. associate-*r* 82.6%

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

      11. *-commutative 82.6%

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

      12. associate-*l* 82.6%

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

   9. Applied egg-rr 82.6%

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

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

   3. Simplified 45.7%

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

   4. Taylor expanded in b around inf 90.7%

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

      1. distribute-lft-out 90.7%

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

   6. Simplified 90.7%

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

      1. *-commutative 97.1%

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

      2. unpow3 97.1%

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

      3. times-frac 97.1%

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

      4. unpow2 97.1%

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

      5. frac-times 97.1%

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

      6. pow1 97.1%

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

      7. metadata-eval 97.1%

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

      8. pow1 97.1%

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

      9. metadata-eval 97.1%

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

      10. pow-sqr 97.1%

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

      11. metadata-eval 97.1%

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

      12. metadata-eval 97.1%

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

   8. Applied egg-rr 90.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 5: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left({b}^{2} - t_0\right)}{-2 \cdot \left(b + \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.01)
     (/ (* (/ 1.0 a) (- (pow b 2.0) t_0)) (* -2.0 (+ b (sqrt t_0))))
     (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.01) {
		tmp = ((1.0 / a) * (pow(b, 2.0) - t_0)) / (-2.0 * (b + sqrt(t_0)));
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.01)
		tmp = Float64(Float64(Float64(1.0 / a) * Float64((b ^ 2.0) - t_0)) / Float64(-2.0 * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $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], -0.01], N[(N[(N[(1.0 / a), $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $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)) < -0.0100000000000000002

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

   3. Simplified 82.1%

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

      1. pow1/2 82.1%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}{a \cdot 2} \]

      2. pow-to-exp 77.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{a \cdot 2} \]

      3. fma-neg 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)} \cdot 0.5}}{a \cdot 2} \]

      4. distribute-lft-neg-in 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      5. *-commutative 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      6. distribute-lft-neg-in 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      7. metadata-eval 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)\right) \cdot 0.5}}{a \cdot 2} \]

      8. associate-*r* 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right) \cdot a}\right)\right) \cdot 0.5}}{a \cdot 2} \]

   5. Applied egg-rr 78.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. frac-2neg 78.0%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)}{-a \cdot 2}} \]

      2. div-inv 77.9%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   7. Applied egg-rr 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-/r* 82.2%

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

      3. flip-- 82.0%

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

      4. frac-times 82.0%

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

      5. pow1 82.0%

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

      6. pow1 82.0%

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

      7. pow-sqr 82.0%

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

      8. metadata-eval 82.0%

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

      9. add-sqr-sqrt 82.7%

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

      10. associate-*r* 82.7%

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

      11. *-commutative 82.7%

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

      12. associate-*l* 82.7%

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

   9. Applied egg-rr 82.7%

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

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

   3. Simplified 45.7%

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

   4. Taylor expanded in b around inf 90.7%

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

      1. distribute-lft-out 90.7%

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

   6. Simplified 90.7%

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

      1. *-commutative 97.1%

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

      2. unpow3 97.1%

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

      3. times-frac 97.1%

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

      4. unpow2 97.1%

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

      5. frac-times 97.1%

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

      6. pow1 97.1%

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

      7. metadata-eval 97.1%

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

      8. pow1 97.1%

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

      9. metadata-eval 97.1%

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

      10. pow-sqr 97.1%

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

      11. metadata-eval 97.1%

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

      12. metadata-eval 97.1%

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

   8. Applied egg-rr 90.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left({b}^{2} - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}{-2 \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 6: 85.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.01)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 2.0))
     (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.01) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(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)) <= -0.01)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $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], -0.01], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $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)) < -0.0100000000000000002

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

   3. Simplified 82.1%

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

      1. pow1/2 82.1%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}}}{a \cdot 2} \]

      2. pow-to-exp 77.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{a \cdot 2} \]

      3. fma-neg 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \color{blue}{\left(\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)\right)} \cdot 0.5}}{a \cdot 2} \]

      4. distribute-lft-neg-in 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot a\right) \cdot c}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      5. *-commutative 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      6. distribute-lft-neg-in 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)\right) \cdot 0.5}}{a \cdot 2} \]

      7. metadata-eval 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, c \cdot \left(\color{blue}{-4} \cdot a\right)\right)\right) \cdot 0.5}}{a \cdot 2} \]

      8. associate-*r* 78.0%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right) \cdot a}\right)\right) \cdot 0.5}}{a \cdot 2} \]

   5. Applied egg-rr 78.0%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. +-commutative 78.0%

         \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5} + \left(-b\right)}}{a \cdot 2} \]

      2. flip-+ 77.9%

         \[\leadsto \frac{\color{blue}{\frac{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5} \cdot e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5} - \left(-b\right) \cdot \left(-b\right)}{e^{\log \left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) \cdot 0.5} - \left(-b\right)}}}{a \cdot 2} \]

   7. Applied egg-rr 82.6%

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

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

   3. Simplified 45.7%

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

   4. Taylor expanded in b around inf 90.7%

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

      1. distribute-lft-out 90.7%

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

   6. Simplified 90.7%

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

      1. *-commutative 97.1%

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

      2. unpow3 97.1%

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

      3. times-frac 97.1%

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

      4. unpow2 97.1%

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

      5. frac-times 97.1%

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

      6. pow1 97.1%

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

      7. metadata-eval 97.1%

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

      8. pow1 97.1%

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

      9. metadata-eval 97.1%

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

      10. pow-sqr 97.1%

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

      11. metadata-eval 97.1%

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

      12. metadata-eval 97.1%

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

   8. Applied egg-rr 90.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 7: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.01)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a b) (pow (/ c b) 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)) <= -0.01) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 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)) <= -0.01)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 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], -0.01], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $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)) < -0.0100000000000000002

   1. Initial program 82.1%

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

   3. Simplified 82.2%

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

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

   3. Simplified 45.7%

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

   4. Taylor expanded in b around inf 90.7%

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

      1. distribute-lft-out 90.7%

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

   6. Simplified 90.7%

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

      1. *-commutative 97.1%

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

      2. unpow3 97.1%

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

      3. times-frac 97.1%

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

      4. unpow2 97.1%

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

      5. frac-times 97.1%

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

      6. pow1 97.1%

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

      7. metadata-eval 97.1%

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

      8. pow1 97.1%

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

      9. metadata-eval 97.1%

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

      10. pow-sqr 97.1%

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

      11. metadata-eval 97.1%

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

      12. metadata-eval 97.1%

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

   8. Applied egg-rr 90.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 8: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.01) t_0 (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0))))))

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 <= -0.01) {
		tmp = t_0;
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.01d0)) then
        tmp = t_0
    else
        tmp = (-c / b) - ((a / b) * ((c / b) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else {
		tmp = (-c / b) - ((a / b) * Math.pow((c / b), 2.0));
	}
	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 <= -0.01:
		tmp = t_0
	else:
		tmp = (-c / b) - ((a / b) * math.pow((c / b), 2.0))
	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 <= -0.01)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	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 <= -0.01)
		tmp = t_0;
	else
		tmp = (-c / b) - ((a / b) * ((c / b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], t$95$0, N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $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)) < -0.0100000000000000002

   1. Initial program 82.1%

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

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 45.7%

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

      1. *-commutative 45.7%

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

   3. Simplified 45.7%

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

   4. Taylor expanded in b around inf 90.7%

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

      1. distribute-lft-out 90.7%

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

   6. Simplified 90.7%

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

      1. *-commutative 97.1%

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

      2. unpow3 97.1%

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

      3. times-frac 97.1%

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

      4. unpow2 97.1%

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

      5. frac-times 97.1%

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

      6. pow1 97.1%

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

      7. metadata-eval 97.1%

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

      8. pow1 97.1%

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

      9. metadata-eval 97.1%

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

      10. pow-sqr 97.1%

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

      11. metadata-eval 97.1%

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

      12. metadata-eval 97.1%

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

   8. Applied egg-rr 90.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 9: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

   1. *-commutative 53.4%

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

3. Simplified 53.4%

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

4. Taylor expanded in b around inf 84.0%

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

   1. distribute-lft-out 84.0%

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

6. Simplified 84.0%

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

   1. *-commutative 92.9%

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

   2. unpow3 92.9%

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

   3. times-frac 92.9%

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

   4. unpow2 92.9%

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

   5. frac-times 92.9%

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

   6. pow1 92.9%

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

   7. metadata-eval 92.9%

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

   8. pow1 92.9%

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

   9. metadata-eval 92.9%

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

   10. pow-sqr 92.9%

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

   11. metadata-eval 92.9%

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

   12. metadata-eval 92.9%

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

8. Applied egg-rr 84.0%

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

9. Final simplification 84.0%

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

Alternative 10: 63.9% 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 53.4%

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

   1. *-commutative 53.4%

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

3. Simplified 53.4%

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

4. Taylor expanded in b around inf 66.3%

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

   1. mul-1-neg 66.3%

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

6. Simplified 66.3%

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

7. Final simplification 66.3%

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

Alternative 11: 1.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(c / 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 53.4%

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

   1. *-commutative 53.4%

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

3. Simplified 53.4%

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

4. Taylor expanded in b around inf 66.2%

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

   1. associate-*r/ 66.2%

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

6. Simplified 66.2%

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

   1. div-inv 66.2%

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

   2. *-commutative 66.2%

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

   3. metadata-eval 66.2%

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

   4. *-commutative 66.2%

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

   5. associate-/r* 66.2%

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

   6. metadata-eval 66.2%

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

   7. metadata-eval 66.2%

      \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \frac{-2 \cdot \left(a \cdot c\right)}{b} \]

   8. associate-*r/ 66.2%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b}\right)} \]

   9. associate-/l* 66.2%

      \[\leadsto \frac{0.5}{a} \cdot \left(-2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right) \]

8. Applied egg-rr 66.2%

   \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(-2 \cdot \frac{a}{\frac{b}{c}}\right)} \]
9. Step-by-step derivation

   1. associate-*l/ 66.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(-2 \cdot \frac{a}{\frac{b}{c}}\right)}{a}} \]

   2. clear-num 66.2%

      \[\leadsto \frac{0.5 \cdot \left(-2 \cdot \color{blue}{\frac{1}{\frac{\frac{b}{c}}{a}}}\right)}{a} \]

   3. associate-/l* 66.2%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{-2 \cdot \frac{1}{\frac{\frac{b}{c}}{a}}}}} \]

   4. clear-num 66.2%

      \[\leadsto \frac{0.5}{\frac{a}{-2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}}} \]

   5. *-commutative 66.2%

      \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\frac{a}{\frac{b}{c}} \cdot -2}}} \]

   6. div-inv 66.2%

      \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\left(a \cdot \frac{1}{\frac{b}{c}}\right)} \cdot -2}} \]

   7. clear-num 66.3%

      \[\leadsto \frac{0.5}{\frac{a}{\left(a \cdot \color{blue}{\frac{c}{b}}\right) \cdot -2}} \]

   8. *-commutative 66.3%

      \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\left(\frac{c}{b} \cdot a\right)} \cdot -2}} \]

   9. associate-*l* 66.3%

      \[\leadsto \frac{0.5}{\frac{a}{\color{blue}{\frac{c}{b} \cdot \left(a \cdot -2\right)}}} \]

   10. add-sqr-sqrt 0.0%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \color{blue}{\left(\sqrt{a \cdot -2} \cdot \sqrt{a \cdot -2}\right)}}} \]

   11. sqrt-unprod 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \color{blue}{\sqrt{\left(a \cdot -2\right) \cdot \left(a \cdot -2\right)}}}} \]

   12. swap-sqr 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \sqrt{\color{blue}{\left(a \cdot a\right) \cdot \left(-2 \cdot -2\right)}}}} \]

   13. unpow2 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \sqrt{\color{blue}{{a}^{2}} \cdot \left(-2 \cdot -2\right)}}} \]

   14. metadata-eval 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \sqrt{{a}^{2} \cdot \color{blue}{4}}}} \]

   15. metadata-eval 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \sqrt{{a}^{2} \cdot \color{blue}{{2}^{2}}}}} \]

   16. unpow-prod-down 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \sqrt{\color{blue}{{\left(a \cdot 2\right)}^{2}}}}} \]

   17. unpow2 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \sqrt{\color{blue}{\left(a \cdot 2\right) \cdot \left(a \cdot 2\right)}}}} \]

   18. sqrt-prod 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \color{blue}{\left(\sqrt{a \cdot 2} \cdot \sqrt{a \cdot 2}\right)}}} \]

   19. add-sqr-sqrt 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \color{blue}{\left(a \cdot 2\right)}}} \]

   20. *-commutative 1.6%

       \[\leadsto \frac{0.5}{\frac{a}{\frac{c}{b} \cdot \color{blue}{\left(2 \cdot a\right)}}} \]

10. Applied egg-rr 1.6%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\frac{c}{b} \cdot \left(2 \cdot a\right)}}} \]
11. Step-by-step derivation

    1. associate-/r* 1.6%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{a}{\frac{c}{b}}}{2 \cdot a}}} \]

    2. associate-/l* 1.6%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{a \cdot b}{c}}}{2 \cdot a}} \]

    3. associate-*r/ 1.6%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{a \cdot \frac{b}{c}}}{2 \cdot a}} \]

    4. *-commutative 1.6%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{b}{c} \cdot a}}{2 \cdot a}} \]

    5. associate-/l* 1.6%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 \cdot a\right)}{\frac{b}{c} \cdot a}} \]

    6. associate-*r* 1.6%

       \[\leadsto \frac{\color{blue}{\left(0.5 \cdot 2\right) \cdot a}}{\frac{b}{c} \cdot a} \]

    7. metadata-eval 1.6%

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

    8. *-lft-identity 1.6%

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

    9. associate-/l/ 1.6%

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

    10. *-inverses 1.6%

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

    11. associate-/r/ 1.6%

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

    12. associate-*l/ 1.6%

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

    13. associate-*r/ 1.6%

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

    14. *-lft-identity 1.6%

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

12. Simplified 1.6%

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

13. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023336 
(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)))